Parabola is a fundamental concept in mathematics and geometry. It is one of the conic sections in Maths which is formed by an intersection of a surface plane and a double-napped cone. A parabola is a U-shaped curve that can be either concave up or down, depending on the equation. This U-shaped curve appears in various real-world applications, from the paths of projectiles in physics to the design of satellite dishes and suspension bridges. Parabolic curves are widely used in many fields such as physics, engineering, finance, and computer sciences.
In this article, we will understand What a Parabola is, its Graph, Properties, Examples and the Equation of Parabola, and in detail.
What is Parabola ?
Parabola is an equation of a specific curve, such that each point on the curve is always equidistant from a fixed point and a fixed-line. Parabola is the introductory curves in the study of conic section, as parabola and other conics can be obtained by slicing the double-napped cone (two identical cones stacked on top of each other from the vertex part) with the help of the surface plane at different angles.
Parabola Definition
Parabola is a symmetrical curve which is defined as the set of all points that are equidistant from a fixed point (called the focus) and a fixed straight line (called the directrix). The focus and directrix are two essential elements that determine the shape and position of a parabola.
Parabola Shape
A parabola is a U-shaped curved line where every point on the line is at an equal distance from the focus and directrix of the parabola.
Parabola
Equation of Parabola
Equation of Parabola can vary depending on its orientation and the position of its vertex, but one common form is:
y = ax2 + bx + c
Here, a, b, and c are constants. The shape of the parabola depends primarily on the value of a:
- If a > 0, the parabola opens upwards.
- If a < 0, the parabola opens downwards.
Standard Equation of Parabola
Standard Equation of Parabola is given as follows:
y2 = 4ax
In this form, directrix is parallel to the y-axis.
If directrix is parallel to the x-axis, then the standard equation of a parabola is given by,
x2 = 4ay
If the parabolas are drawn in alternate quadrants, then their equation is given as y2 = -4ax and x2 = -4ay.
|
Form
|
y2 = 4ax
|
y2 = – 4ax
|
x2 = 4ay
|
x2 = – 4ay
|
| Vertex |
(0, 0)
|
(0,0)
|
(0, 0)
|
(0, 0)
|
| Focus |
(a, 0)
|
(-a, 0)
|
(0, a)
|
(0, -a)
|
| Equation of Axis |
y = 0
|
y = 0
|
x = 0
|
x = 0
|
| Equation of Directrix |
x = – a
|
x = a
|
y = – a
|
y = a
|
| Length of Latus Rectum |
4a
|
4a
|
4a
|
4a
|
General equation of a parabola is given by y = a(x – h)2 + k or x = a(y – k)2 +h where (h, k) denotes the vertex.
(Regular form) y = a(x – h)2 + k
(Sidewise from) x = a(y – k)2 + h
Parametric Coordinates of a Parabola
For a parabola, y2 = 4ax, if we take x = at2 and y = 2at for any value of “t” they will satisfy the equation of a parabola, the coordinates (at2, 2at) is termed as parametric coordinate, and “t” is called as the parameter.
Thus, x = at2 and y = 2at are called the parametric equations of the parabola y2 = 4ax
Similarly, parametric form of the parabola x2 = 4ay are x = 2at, y = at2
Equation of Tangent to a Parabola
Tangents are lines that touch the curve only at a single point. So a line that touches the parabola exactly at one single point is called the tangent to a parabola.
There are various ways to find the tangent of a parabola which are discussed in next sections.
For the given parabola y2 = 4ax equation of the tangent at point (x1, y1) is given by:
yy1 = 2a(x+x1)
where,
(x1, y1) is the point of contact between the tangent and the curve.
For the given parabola y2 = 4ax equation of the tangent at point (at2, 2at) is given by:
ty = x + at2
where,
(at2, 2at) is the point of contact between the tangent and the curve.
For the given parabola y2 = 4ax with slope m equation of the tangent at point (a/m2, 2a/m) is given by
y = mx + a/m
where,
(a/m2, 2a/m) is the point of contact between the tangent and the curve.
Pair of Tangent from an External Point
Pair of tangents from an external point to any conic is given by SS1 = T2 where for parabola y2 = 4ax, S = y2 – 4ax, S1 = y12 -4ax1 and T = yy1 – 2a(x + x1).
Thus, the equation of pair of tangents from an external point becomes:
(y2 – 4ax)( y12 -4ax1) = [yy1 – 2a(x + x1)]2
Director Circle of Parabola
Director circle is the geometric object related to the conic section and is defined as the locus of the intersection of the pair perpendicular tangent of any conic. For the parabola, the director circle is the directrix as all the perpendicular pairs of tangents of the parabola intersect each other at the directrix.
Chord of contact of the parabola is a secant line joining the point of tangency for the tangents drawn from the external point on the parabola. For parabola y2 = 4ax, chord of contact is given by T = 0, where T = yy1 – 2a(x + x1). Therefore the equation of chord of contact is given
T = yy1 – 2a(x + x1) =0
Where, (x1, y1) is the external point from which both the tangents are drawn to the parabola.
Equation of Normal to a Parabola
A line perpendicular to the tangent of the parabola at the point of tangency is known as the normal of the parabola. As this line is perpendicular to the tangent at the point of tangency to the parabola, the equation of this line can be found easily if the equation of tangent and point of tangency is given, using the concept of the equation of line perpendicular to the given line, but this is not always the case.
The equation of normal is given in three ways as follows:
For a parabola y2 = 4ax and m is the slope of normal at the point of contact (am2, -2am), the equation of normal is given by:
y = mx – 2am – am3
For a parabola y2 = 4ax, equation of normal at (x1, y1) is given as follows:
y – y1 = (-y1/2a)(x – x1)
For a parabola y2 = 4ax, the equation of normal at the point (at2, 2at) [where t is the parameter] is given as follows:
y = -tx + 2at + at3
Some important parabola formulas are added in the table below:
| Formula/Property |
Equation or Description |
| Standard Equation of Parabola |
y2 = 4ax (or x2 = 4ay for a different orientation) |
| Focus |
(a, 0) for y2 = 4ax (or (0, a) for x2 = 4ay) |
| Directrix |
x = -a for y2 = 4ax (or y = -a for x2 = 4ay) |
| Latus Rectum |
4a |
| Eccentricity |
1 |
| General Equations of Parabola |
y = a(x – h)2 + k OR x = a(y – k)2 + h |
| Parametric Equation of Parabola |
x = 2at, y = at2 |
| Equation of Tangent |
yy1 = 2a(x + x1) |
| Equation of Tangent in Parametric Form |
ty = x + at2 [where, (at2, 2at) is the point of contact] |
| Equation of Tangent in Slope Form |
y = mx + a/m [where m is the slope of tangent] |
| Pair of Tangent from an External Point |
(y2 – 4ax)( y12 -4ax1) = [yy1 – 2a(x + x1)]2 |
| Director Circle |
Directrix i.e., x = -a [For y2 = 4ax] |
| Chord of Contact |
yy1 – 2a(x + x1) =0 |
| Equation of Normal in Slope Form |
y = mx – 2am – am3 |
| Equation of Normal in Normal Form |
y – y1 = (-y1/2a)(x – x1) |
| Equation of Normal in Parametric Form |
y = -tx + 2at + at3 |
Articles Related to Parabola:
Derivation of Parabola Equation
Take a point P with coordinates (x, y) on the parabola which lies on the X-Y plane. By the definition of the parabola, the distance of any point on the parabola from the focus and from the directrix is equal.
Now distance of P from the directrix is given by PB where the coordinates of B are (-a, y) as it lies on the directrix, and the distance of P from focus is PF.
Image of parabola is shown below,
By the definition of parabola, PF = PB . . . .(1)
Using Distance Formula, we get
PF = √(x-a)2+(y-0)2= √{(x-a)2+y2} . . . .(2)
PB = √{(x+a)2} . . . .(3)
By using, equations (1), (2), and (3), we get
√{(x-a)2+y2} = √{(x+a)2}
⇒ (x – a)2 + y2 = (x + a)2
⇒ x2 + a2 – 2ax + y2 = x2 + a2 + 2ax
⇒ y2 – 2ax = 2ax
y2 = 4ax
Which is the required equation of the parabola.
Similarly, the equation for other parabolas i.e., x2 = 4ay, y2 = -4ax, and x2 = -4ay, can also be proved.
Graph of Parabola
Graph of the parabola is a U-shaped curve, which can open either in an upward direction or in a downward direction. Generally, the equation of a parabola which is graphed is written in the form of y = ax2 + bx + c, where a, b, and c are constants that define the shape of the parabola.
If a > 0, in the above equation, the parabola opens in an upward direction and its vertex is the lowest point of the parabola, and if a<0, then the parabola opens in a downward direction and its vertex is the highest point in the parabola. Vertex of the parabola is also the point from where the only line of symmetry of the parabola passes.
Position of Point Relative to the Parabola
Position of a point A (x1, y1) relative to the parabola y2 = 4ax, can be shown using the S1 = y2 – 4ax,
Case 1: If S1 = 0, for any point A, then Point A lies on the parabola.
Case 2: If S1 < 0, for any point A, then Point A lies inside the parabola.
Case 3: If S1 > 0, for any point A, then Point A lies outside the parabola.
Intersection with Straight Line
For a parabola y2 = 4ax, any straight-line y = mx + c, can almost intersect the parabola at two points. For the intersection of the line and parabola, put y = mx + c in the equation of the parabola,
(mx + c)2 = 4ax
⇒ m2x2 + c2 + 2mxc = 4ax
⇒ m2x2 + (2mc – 4a)x + c2= 0
⇒ Discriminant = (2mc – 4a)2 – 4m2c2
Case 1: Discriminant > 0:
For positive discriminant, quadratic equations have two real solution,
⇒ There are two points of intersection between line and parabola.
Case 2: Discriminant = 0:
For discriminant equal to 0, quadratic equations have only real solution (common root),
⇒ There are open point of intersection between line and parabola i.e., line is tangent to parabola.
Case 3: Discriminant < 0:
For negative discriminant, quadratic equations have no real roots,
⇒ There are no point of intersection between line and parabola
Properties of Parabola
- Axis of Symmetry: Line that is perpendicular to the directrix and passes through the focus is called the axis of symmetry. The parabola is symmetrical about its axis of symmetry.
- Vertex: Point where the parabola intersects its axis of symmetry is called the vertex. The vertex is the point where the parabola changes direction from opening upwards to opening downwards (or vice versa).
- Focal Length: Distance between the vertex and the focus is called the focal length. All parabolas with the same focal length are similar.
- Directrix: Fixed line from which any point on the parabola is the same distance as the focus.
- Reflective Property: If a parabola is made of a material that reflects light, then light that travels parallel to the axis of symmetry of a parabola and strikes its concave side is reflected to its focus, regardless of where on the parabola the reflection occurs. Conversely, light that originates from a point source at the focus is reflected into a parallel (“collimated”) beam, leaving the parabola parallel to the axis of symmetry.
- Equation of Parabola: Equation of a parabola depends on its orientation and position. The standard equation of a parabola that opens upwards and is centered at the origin is y = x2. The standard equation of a parabola that opens downwards and is centered at the origin is y = -x2.
- Focus: Point (a, 0) is called the focus of a Standard Parabola (y2 = 4ax), and it has a very special property that has various real-life applications i.e. if any light ray traveling parallel to the axis of the parabola, the parabola converges those light rays at the focus.
- Directrix: A line drawn perpendicular to the base axis and passing through the point (-a, 0) is the directrix of the parabola. Directrix is always perpendicular to the axis of a parabola.
- Focal Chord: A chord passing through the focus of the parabola is called the focal chord of a parabola. The focal chord always cuts the parabola at two distinct points.
- Focal Distance: Distance of any point (x1,y1) lying on the parabola, from the focus of the parabola, is called the focal distance. The focal distance equals the perpendicular distance of the given point from the directrix.
- Latus Rectum: A chord perpendicular to the axis of the parabola and passing through the focus of the parabola is called the Latus rectum. Points at the ending of the latus rectum are (a, 2a), (a, -2a), and their length is taken as LL’ = 4a.
- Eccentricity: Ratio of the distance of a point from the focus, and the distance of the point from the directrix is called eccentricity. Eccentricity for a parabola is 1.
Solved Examples on Parabola
Example 1: Find coordinates of the focus, axis, the equation of the directrix, and latus rectum of the parabola y2 = 16x.
Solution:
Given equation of the parabola is: y2 = 16x
Comparing with the standard form y2 = 4ax,
4a = 16 ⇒ a = 4
The coefficient of x is positive so the parabola opens to the right.
Also, the axis of symmetry is along the positive x-axis.
Therefore,
Focus of the parabola is (a, 0) = (4, 0).
Equation of the directrix is x = -a, i.e. x = -4
Length of the latus rectum = 4a = 4(4) = 16
Example 2: Find the equation of the parabola which is symmetric about the y-axis, and passes through the point (3, -4).
Solution:
Given that parabola is symmetric about the y-axis and has its vertex at the origin.
Thus, equation can be of the form x2 = 4ay or x2 = -4ay, where the sign depends on whether the parabola opens upwards or downwards.
Since parabola passes through (3, -4) which lies in the fourth quadrant, it must open downwards.
So, equation will be: x2 = -4ay
Substituting (3, -4) in the above equation,
(3)2 = -4a(-4)
9 = 16a
a = 9/16
Hence, the equation of the parabola is: x2 = -4(9/16)y
4x2 = -9y
Example 3: Find coordinates of the focus, axis, and the equation of the directrix and latus rectum of the parabola y2 = 8x.
Solution:
Given equation of the parabola is: y2 = 8x
Comparing with the standard form y2 = 4ax,
4a = 8
a = 2
The coefficient of x is positive so the parabola opens to the right.
Also, the axis of symmetry is along the positive x-axis.
Therefore,
Focus of the parabola is (a, 0) = (2, 0).
Equation of the directrix is x = -a, i.e. x = -2
Length of the latus rectum = 4a = 4(2) = 8
Example 4: Find the coordinates of the focus, axis, the equation of the directrix, and latus rectum of the parabola y2 = 52x.
Solution:
Given equation of parabola is: y2 = 52x
Comparing with the standard form y2 = 4ax,
4a = 52
a = 13
The coefficient of x is positive so the parabola opens to the right.
Also, the axis of symmetry is along the positive x-axis.
Therefore,
Focus of the parabola is (a, 0) = (13, 0).
Equation of the directrix is x = -a, i.e. x = -13
Length of the latus rectum = 4a = 4(13) = 52
Example 5: Find coordinates of the focus, axis, the equation of the directrix, and latus rectum of the parabola x2 = 16y.
Solution:
Given equation of parabola is: x2 = 16y
Comparing with standard form x2 = 4ay,
4a = 16
a = 4
The coefficient of x is positive so the parabola opens upward.
Also, the axis of symmetry is along the positive x-axis.
Therefore,
Focus of the parabola is (0,a) = (0, 4).
Equation of the directrix is y= -a, i.e. y = -4
Length of the latus rectum = 4a = 4(4) = 16
Practice Questions on Parabola
Q1. Find the vertex, focus, and directrix of the parabola with the equation y = x2 – 4x + 3y = x2 – 4x + 3.
Q2. Determine whether the parabola with the equation y = -2x2 + 4x – 1y = -2x2 + 4x – 1 opens upward or downward, and find its vertex.
Q3. Given the equation 4x2 – 16y = 0, 4x2 – 16y = 0, rewrite it in standard form and find the vertex, focus, and directrix of the parabola.
Q4. Solve for xx in the equation 2x2 – 3x – 5 = 0, 2x2 – 3x – 5 = 0, and determine the nature of the roots with respect to the corresponding parabola.
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Conclusion
Understanding the properties and applications of parabola is essential in both theoretical and applied mathematics. Their unique geometric characteristics and reflective properties make them a fascinating topic with wide-ranging applications in various fields. Whether grappling with parabola motion, designing architectural elements, or studying the nature of conic sections, the parabola remains a key concept.
A parabola is a fundamental concept in mathematics, particularly in the study of quadratic functions and conic sections. Its distinct U-shaped curve is defined by a quadratic equation and exhibits unique properties such as symmetry and a focal point, which have wide-ranging applications in various fields.
Parabola – FAQs
What is Parabola?
Parabola is the set of all points that are equidistant from a fixed point (called the focus) and a fixed straight line (called the directrix).
What is meant by Conjugate Axis of a Parabola?
A line that passes through the vertex of the parabola and is perpendicular to its transverse axis is called the conjugate axis of the parabola.
What are Applications of Parabola?
Parabolas are used for a variety of purposes some of them are
- Parabolic arch is used in the construction of various monuments.
- Parabolic mirrors are used in reflecting telescopes, satellites, etc.
- Parabolas are used in various mathematical calculations, such as tracing the path of a missile, the trajectory of a bullet, etc.
What is Shape of a Parabola?
Graph of a parabola is in the shape of U.
What is Eccentricity of a Parabola?
Eccentricity is defined as the ratio of distances of any point of conic section to its focus and corresponding directrix. For parabola eccentricity is 1.
For parabola y2 = 4ax, length of the latus rectum is calculated by 4a..
What is Vertex of a Parabola?
Point of intersection of the parabola and both the conjugate axis is called the vertex of a parabola. If the equation of parabola is y2 = 4ax, then the vertex is (0, 0).
What is a Parabolic motion?
Parabola motion, also known as projectile motion, describes the path followed by an object that is subject only to the force of gravity and no other external forces, such as air resistance. In this type of motion, the object moves in a curved path that resembles a parabola.
What are 4 Types of Parabola?
Parabolas can generally be categorized into four types based on their orientation and direction:
- Vertical Opening Up: Standard form is y = ax2 + bx + c, where a > 0.
- Vertical Opening Down: Standard form is y = ax2+ bx + c, where a < 0.
- Horizontal Opening Right: Standard form is x = ay2+ by + c, where a > 0.
- Horizontal Opening Left: Standard form is x = ay2+ by + c, where a < 0.
What is difference between Parabola and Hyperbola?
Parabola is a U-shaped curve where any point is at an equal distance from a fixed point (the focus) and a fixed straight line (the directrix). It has one axis of symmetry.
Hyperbola is a curve where the difference of the distances from two points (the foci) is constant. It has two disconnected parts and two axes of symmetry.
What are 4 key Features of a Parabola?
- Vertex: Highest or lowest point of the parabola.
- Axis of Symmetry: Line that divides the parabola into two symmetric halves.
- Focus: A fixed point through which all the parallel light rays reflect off a parabolic mirror.
- Directrix: A fixed line that is equidistant from all points on the parabola.
Who invented Parabola?
Concept of the parabola and its mathematical properties have been studied and developed over centuries by various mathematicians and scholars. However, it was the ancient Greek mathematician Apollonius of Perga who extensively studied conic sections, including the parabola, around the 3rd century BCE.
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A square is a type of quadrilateral where all four sides are of equal length and each interior angle measures 90°. It has two pairs of parallel sides, with opposite sides being parallel. The diagonals of a square are equal in length and bisect each other at right angles.Squares are used in various f
6 min read
Rectangle | Definition, Properties, Formulas
A rectangle is a quadrilateral with four sides and following properties. All four angles are right angles (90°). The opposite sides of a rectangle are equal in length and parallel to each other.A rectangle is a two-dimensional flat shape. Here, sides AB and CD are equal and parallel and are called t
8 min read
Trapezium in Maths | Formulas, Properties & Examples
A trapezium or Trapezoid is a four-sided quadrilateral with one pair of parallel sides, called the bases, and the other two non-parallel sides called the legs. The term "trapezium" comes from the Greek word "trapeze," meaning "table."It is a two-dimensional shape with four sides and four vertices. T
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Kite - Quadrilaterals
A Kite is a special type of quadrilateral that is easily recognizable by its unique shape, resembling the traditional toy flown on a string. In geometry, a kite has two pairs of adjacent sides that are of equal length. This distinctive feature sets it apart from other quadrilaterals like squares, re
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Area of Parallelogram | Definition, Formulas & Examples
A parallelogram is a four-sided polygon (quadrilateral) where opposite sides are parallel and equal in length. In a parallelogram, the opposite angles are also equal, and the diagonals bisect each other (they cut each other into two equal parts). The area of a Parallelogram is the space or the regio
9 min read
Circle
Circles in Maths
A circle is a two-dimensional shape where all points on the circumference are the same distance from the center. A circle consists of all points in a plane that are equidistant (at the same distance) from a fixed point called the centre. The distance from the centre to any point on the circle is cal
10 min read
Circumference of Circle - Definition, Perimeter Formula, and Examples
The circumference of a circle is the distance around its boundary, much like the perimeter of any other shape. It is a key concept in geometry, particularly when dealing with circles in real-world applications such as measuring the distance traveled by wheels or calculating the boundary of round obj
7 min read
Area of a Circle: Formula, Derivation, Examples
The area of a Circle is the measure of the two-dimensional space enclosed within its boundaries. It is mostly calculated by the size of the circle's radius which is the distance from the center of the circle to any point on its edge. The area of a circle is proportional to the radius of the circle.
10 min read
Area of a Circular Sector
A circular sector or circle sector is the portion of a disk enclosed by two radii and an arc, where the smaller area is known as the minor sector and the larger being the major sector. Let's look at this figure and try to figure out the sector: In this figure the green shaded part is a sector, "r" i
4 min read
Segment of a Circle
Segment of a Circle is one of the important parts of the circle other than the sector. As we know, the circle is a 2-D shape in which points are equidistant from the point and the line connecting the two points lying on the circumference of the circle is called the chord of the circle. The area form
7 min read
Circle Theorems
Circle is a collection of points that are at a fixed distance from a particular point. The fixed point is called the centre of the circle and the fixed distance is called the radius of the circle. We come across many objects in real life which are round in shape. For example, wheels of vehicles, ban
5 min read
Tangent to a Circle
Tangent in Circles are the line segments that touch the given curve only at one particular point. Tangent is a Greek word meaning "To Touch". For a circle, we can say that the line which touches the circle from the outside at one single point on the circumference is called the tangent of the circle.
10 min read
Theorem - The tangent at any point of a circle is perpendicular to the radius through the point of contact - Circles | Class 10 Maths
Tangent is a straight line drawn from an external point that touches a circle at exactly one point on the circumference of the circle. There can be an infinite number of tangents of a circle. These tangents follow certain properties that can be used as identities to perform mathematical computations
3 min read
Number of Tangents from a Point on a Circle
A circle is a collection of all the points in a plane that are at a constant distance from a particular point. This distance is called the radius of the circle and the fixed point is called the centre. Â A straight line and a circle can co-exist in three ways, one can be a straight line with no inter
12 min read
Theorem - The lengths of tangents drawn from an external point to a circle are equal - Circles | Class 10 Maths
Tangent is a straight line drawn from an external point that touches a circle at exactly one point on the circumference of the circle. There can be an infinite number of tangents of a circle. These tangents follow certain properties that can be used as identities to perform mathematical computations
5 min read
Equation of a Circle
A circle is a geometric shape described as the set of all points in a plane that are equidistant from a fixed point called the center. The distance from the center to any point on the circle is called the radius. Some key components of the circle are: Center: The fixed point in the middle of the cir
15 min read
What is Cyclic Quadrilateral
Cyclic Quadrilateral is a special type of quadrilateral in which all the vertices of the quadrilateral lie on the circumference of a circle. In other words, if you draw a quadrilateral and then find a circle that passes through all four vertices of that quadrilateral, then that quadrilateral is call
9 min read
The sum of opposite angles of a cyclic quadrilateral is 180° | Class 9 Maths Theorem
In Euclidean geometry, a cyclic quadrilateral or inscribed quadrilateral is a quadrilateral whose vertices all lie on a single circle. This circle is called the circumcircle or circumscribed circle, and the vertices are said to be concyclic. The center of the circle and its radius are called the cir
6 min read
3D Geometry
Visualizing Solid Shapes
Visualizing Solid Shapes: Any plane or any shape has two measurements length and width, which is why it is called a two-dimensional(2D) object. Circles, squares, triangles, rectangles, trapeziums, etc. are 2-D shapes. If an object has length, width, and breadth then it is a three-dimensional object(
8 min read
Polyhedron | Meaning, Shapes, Formula, and Examples
A polyhedron is a 3D solid made up of flat polygonal faces, with edges meeting at vertices. Each face is a polygon, and the edges connect the faces at their vertices. Examples include cubes, prisms, and pyramids. Shapes like cones and spheres are not polyhedrons because they lack polygonal faces. Po
7 min read
Difference between 2D and 3D Shapes
2D shapes are flat like pictures on paper, with just length and breadth but not depth. On the other hand, 3D shapes are like real objects you can touch, with length, breadth, and depth. They take up space, like a toy that you can hold. Examples of 2D shapes include squares and circles. Cubes, sphere
3 min read
Lines
Equation of a Straight Line | Forms, Examples and Practice Questions
The equation of a line describes the relationship between the x-coordinates and y-coordinates of all points that lie on the line. It provides a way to mathematically represent that straight path. In general, the equation of a straight line can be written in several forms, depending on the informatio
11 min read
Slope of a Line
Slope of a Line is the measure of the steepness of a line a surface or a curve whichever is the point of consideration. The slope of a Line is a fundamental concept in the stream of calculus or coordinate geometry or we can say the slope of a line is fundamental to the complete mathematics subject.
12 min read
Angle between a Pair of Lines
Given two integers M1 and M2 representing the slope of two lines intersecting at a point, the task is to find the angle between these two lines. Examples: Input: M1 = 1.75, M2 = 0.27Output: 45.1455 degrees Input: M1 = 0.5, M2 = 1.75Output: 33.6901 degrees Approach: If ? is the angle between the two
4 min read
Slope Intercept Form
The slope-intercept formula is one of the formulas used to find the equation of a line. The slope-intercept formula of a line with slope m and y-intercept b is, y = mx + b. Here (x, y) is any point on the line. It represents a straight line that cuts both axes. Slope intercept form of the equation i
9 min read
Point Slope Form Formula of a Line
In geometry, there are several forms to represent the equation of a straight line on the two-dimensional coordinate plane. There can be infinite lines with a given slope, but when we specify that the line passes through a given point then we get a unique straight line. Different forms of equations o
6 min read
Writing Slope-Intercept Equations
Straight-line equations, also known as "linear" equations, have simple variable expressions with no exponents and graph as straight lines. A straight-line equation is one that has only two variables: x and y, rather than variables like y2 or √x. Because it contains information about these two proper
10 min read
Slope of perpendicular to line
You are given the slope of one line (m1) and you have to find the slope of another line which is perpendicular to the given line. Examples: Input : 5 Output : Slope of perpendicular line is : -0.20 Input : 4 Output : Slope of perpendicular line is : -0.25 Suppose we are given two perpendicular line
3 min read
What is the Point of Intersection of Two Lines Formula?
Point of Intersection of Two Lines Formula: The point of intersection is the point where two lines or two curves meet each other. The point of intersection of two lines or two curves is a point. If two planes meet each other then the point of intersection is a line. It is defined as the common point
5 min read
Slope of the line parallel to the line with the given slope
Given an integer m which is the slope of a line, the task is to find the slope of the line which is parallel to the given line. Examples: Input: m = 2 Output: 2 Input: m = -3 Output: -3 Approach: Let P and Q be two parallel lines with equations y = m1x + b1, and y = m2x + b2 respectively. Here m1 an
3 min read
Minimum distance from a point to the line segment using Vectors
Given the coordinates of two endpoints A(x1, y1), B(x2, y2) of the line segment and coordinates of a point E(x, y); the task is to find the minimum distance from the point to line segment formed with the given coordinates.Note that both the ends of a line can go to infinity i.e. a line has no ending
10 min read
Distance between two parallel lines
Given are two parallel straight lines with slope m, and different y-intercepts b1 & b2.The task is to find the distance between these two parallel lines.Examples: Input: m = 2, b1 = 4, b2 = 3 Output: 0.333333 Input: m = -4, b1 = 11, b2 = 23 Output: 0.8 Approach: Let PQ and RS be the parallel lin
4 min read
Equation of a straight line passing through a point and making a given angle with a given line
Given four integers a, b, c representing coefficients of a straight line with equation (ax + by + c = 0), the task is to find the equations of the two straight lines passing through a given point [Tex](x1, y1) [/Tex]and making an angle ? with the given straight line. Examples: Input: a = 2, b = 3, c
15+ min read