Tangent and Normals are the lines that are used to define various properties of the curves. We define tangent as the line which touches the circle only at one point and normal is the line that is perpendicular to the tangent at the point of tangency. Any tangent of the curve passing through the point (x1, y1) is of the form y – y1 = m(x-x1) and the equation of the normal at that point is represented using y – y1 = -1/m(x-x1) where m is the slope of the line.
Tangents and normals are lines related to curves. A tangent is a line that touches a curve at a specific point without crossing it at that point, and every point on a curve has its tangent. A normal, on the other hand, is a line that is perpendicular to the tangent at the point where the tangent contacts the curve.
Let us learn more about the equation of tangents and normals for various curves like circles, parabolas, and other curves, examples, and others in this article.
What are Tangents and Normals?
Tangents and Normals are the lines that we associate with any curve that help us to study and solve various problems of the curves. A line that touches the curve at only one point is the tangent of the curve and any line perpendicular to the curve at the point of tangency is the normal to that curve at that point.
We can draw various tangents to the curve which touches the curve at only one point and the normal can be drawn to the curve at that point. As tangent and normal are straight lines so we represent them using the equation of the straight line. We can easily draw tangent and normal to all types of the curve such as Circles, Ellipses, Parabolas, Hyperbolas, and others. Now let’s learn about Tangent and normal in detail.
What are Tangents?
Tangent is a line that touches a curve at a particular point externally. At a given point of a curve, one and only one tangent passes through it. For a given curve tangent at a particular point is a straight line that just touches that curve at that point, and it goes in the same direction as the curve at that point. We can assume tangent as a line that touches the circle only once. For example, If we take a stone tied to a rope and rotate the stone then at any point if we stop rotating the stone then it goes in a straight line, which is called the tangent to the line at that point.
Tangent Definition
A tangent is a line or plane that touches a curve or curved surface at exactly one point and does not intersect it at or near that point. This line or plane is said to be “tangent” to the curve at that specific point.
For any curve, F(x) the tangent at point (x1, y1) is defined at the line, y – y1 = m(x-x1) where m is the slope of the line.

Tangents and Normals
Properties of Tangents
The tangent to a curve has various properties and some of them are,
- Tangents touch the curve only at one point.
- If any tangent to a curve y = f(x) makes angle θ with the x-axis, then dy/dx = Slope of Tangent = tan θ.
- If the slope of the tangent is zero, then tan θ will be equal to 0 and so θ = 0 which implies that the tangent line is parallel to the x-axis.
- If the slope of the tangent is (θ) = π/2, then tan θ will approach ∞, i.e., the tangent line is perpendicular to the x-axis.
What are Normal?
Normal to the curve at any given point is defined as the line passing through the curve which is perpendicular to the tangent of the curve at the point of tangency.
Normal Definition
Normal refers to a line that is perpendicular to the tangent line at the point of tangency on the curve. This perpendicular line extends in both directions from the point where it meets the curve.
For any curve, F(x) the equation of normal at point (x1, y1) is defined at the line, y – y1 = -1/m(x-x1) where m is the slope of the tangent line passing through that point.

Tangents and Normals
Properties of Normals
The normal line to the curve has various properties and some of them are,
- A normal line at any point of a circle will always pass through the center of the circle.
- The normal to any curve is always perpendicular to the tangent at any point on the curve.
How To Find Tangents and Normals?
To find the Tangents and Normals to a curve we require the equation of the curve and the point at which we have to find the tangent and normal. Suppose the point at which we require to find the equation of the Tangents and Normals to the curve is (x1, y1) and the equation of the curve is f(x) then we can easily find the equation of the tangent and normal to the curve.
We know that the tangents and the normal are perpendicular to each other and the slope of the curve y = f(x) at any point (x1, y1) is given using the formula,
m = (dy/dx) at (x1, y1)
We also know that if the slope of the tangent line is m1 and the slope of the perpendicular line is m2 then,
m1 × m2 = -1
Using these we can easily find the tangent and normal to any curve of the circle.
Equation of Tangent and Normal to the Curve
We can calculate the equation of tangent and normal of the curve by various means that include.
- In Cartesian Coordinates System
- In Parametric Form
In Cartesian Coordinates System
At a point on the curve, the gradient of the curve is equal to the gradient of the tangent to the curve at that point. So, the equation of a tangent can be found by the gradient at that point to the curve and the given point as follows,
As we know that the equation of the straight line passes through point P (x0, y0) is
y – y0 = m(x – x0)
Here, m is the finite slope of the line. Now the slope of the tangent to a curve given is y = f(x) at point P (x0, y0) is f'(x0). Then the equation of the tangent to the curve at point P(x0, y0) is
y – y0 = f'(x0)(x – x0)
For the normal, as we already know that the normal is always perpendicular to the tangent line. Then the slope of the normal to the curve will be:
Slope of Normal = -1/f'(x0)
So, the equation of normal to the curve y = f(x) at the point (x0, y0) is,
y – y0 = [-1/f'(x0)](x – x0)
f'(x0)(y – y0) + (x – x0) = 0
Let us assume the parametric form of the curve is
x = x(t) ….(i)
y = y(t) ….(ii)
Now we find the slope of the tangent to a curve at the point (x0, y0), by using the differentiation rule:
m = tan α = y'(t)/x'(t)
Hence, the equation of the tangent is:
y – y0 = [y'(t)/x'(t)](x – x0)
Accordingly, the equation of the normal is:
y – y0 = – x'(t)/y'(t)(x – x0)
y'(t)(y – y0) + x'(t)(x – x0) = 0
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Tangents and Normals for Various Curves
We can easily find the tangent and normal to various curves and some of the important curves and the equation of their tangent and normal are,
Circle: For the circle represented using the equation x2 + y2 + 2gx + 2fy + c = 0 and at the point (x1, y1). The equations for Tangent and Normal for the circle are given as,
Tangent: xx1 + yy1 + g(x + x1) + f(y + y1) + c = 0
Normal: (y – y1)/(y1 + f) = (x – x1)/(x1 + g)
Parabola: For the parabola represented using the equation y2 = 4ax, and at the point (x1, y1). The equations for Tangent and Normal for parabola are given as,
Tangent: yy1 = 2a(x + x1)
Normal: (y-y1) = (-y1/2a)(x – x1)
Ellipse: For the ellipse represented using the equation x2/a2 + y2/b2 = 1, and at the point (x1, y1). The equations for Tangent and Normal for ellipse are given as,
Tangent: xx1/a2 + yy1/b2 = 1
Normal: a2x/x1 – b2y/y1 = a2 – b2
Hyperbola: For the hyperbola represented using the equation x2/a2 – y2/b2 = 1, and at the point (x1, y1). The equations for Tangent and Normal for hyperbola are given as,
Tangent: xx1/a2 – yy1/b2 = 1
Normal: (y – y1) = (-a2y1/b2x1)(x – x1)
Similarly the equation of the normal at the point (x1, y1), for each of the above-given curves, can be calculated by taking the negative of the inverse differentiation of the curve at the given point, as the slope of the curve and then forming the equation of the normal using the formula for the equation of the line passing through the given point and having slope m.
Important Maths Related Links:
Practice Problems on Tangents and Normals
1. Find the equation of the tangent line to the curve f(x) = x2 at x = 3.
2. Find the equation of the normal line to the curve g(x) = 3x3 at x = 1.
3. If h(x) = x1/2, calculate the slope of the tangent and the normal at x = 4.
4. Determine the points on the curve y = x2 – 4x + 4 where the tangent is horizontal.
5. Find the equations of both the tangent and the normal to the curve y = x3 – 6x2 + 11x – 6 at x = 2.
Tangents And Normals Examples
Example 1: Find the slope of the tangent and the normal to the curve y = 6x2 – 10x at x = 1.
Solution:
Given Curve y = 6x2 – 10x
dy/dx = 12x − 10
Slope of tangent to the given curve at x = 1 is,
m1 = [dy/dx]x=1 = 12 × 1 – 10 = 2
We know that,
Slope of Tangent × Slope of Normal = m1 × m2 = -1
Slope of Normal = -1/2 = -0.5
Example 2: Find The slope of the tangent and normal to the curve y = 3x3 + 3sin(x) at x = 0.
Solution:
The given curve is y = 3x3 + 3sin(x)
Now the gradient, dy/dx = 9x2 + 3cos(x)
So, the slope of the tangent to the given curve at x = 0 is,
dy/dx]x=0 = 0 + 3 × 1 = 3
The slope of the normal will be:
= -1/3
Example 3: Find the equation of the tangent to the curve y = 6x2 – 2x + 3 at P(1, 0).
Solution:
The given curve is y = 6x2 – 2x + 3
Now the gradient, dy/dx = 12x – 2
So, the slope of the tangent to the given curve at P(1,0) is
dy/dx]1,0 = 12 – 2 = 10
The equation of the tangent will be:
y – 0 = 10(x – 1)
y = 10x – 10
Example 4: Determine the point on the curve y = 6x2 – 8x + 1 where the tangent is parallel to the line y = 4x – 5.
Solution:
Given curve is y = 6x2 – 8x + 1
Now the gradient, dy/dx = 12x – 8
Tangent is parallel to y = 4x – 5
so,
12x0 – 8 = 4
or x0 = 1
Putting x = 1 in equation of the curve
We get,
y0 = 6 – 8 + 1
y0 = -1
So the point is (1, -1)
Example 5: Find the slope of the tangent to the curve given by:
x = psin3u, y = qcos3u at point where u = π/2.
Solution:
Given,
x = psin3u …(i)
y = qcos3u …(ii)
Value of u = π/2
On differentiating eq(i) and (ii), w.r.t u, we get
dx/du = 3psin2u.cosu …(iii)
dy/du = -3qcos2u.sinu …(iv)
Now we find the slope of the tangent at point u = π/2
[Tex]\frac{dy}{dx}=\frac{\frac{dy}{du}}{\frac{dx}{du}} = -\frac{q3cos^2u.sinu}{p3sin^2u.cosu}[/Tex]
dy/dx = -qcosu/psinu
[dy/dx]u=π/2 = -qcos(π/2)/psin(π/2) = 0
Hence, slope of tangent is m = 0
FAQs on Tangents And Normals
What are Tangents?
We define tanget as the curve which touches the curve at only one point and that point is called the point of tangency.
What are Normals?
The normals are the line which are perpendicular to the tangent and are drawn at the point ogf tangency.
How to find Tangents and Normals?
To find tangent and normal we require the point (say x1, y1) on the curve at which the tangent is required and then we find the differentiation of the curve at that point which gives the slope of the tangent at that point, (say m). Now if the slope of the line m and the point at which the line passes (x1, y1) is given then we find the equation of the tangent by,
y – y1 = m(x – x1)
For the equation of normal, we take its slope as -1/m then find the equation of the line using the above formula.
What is the Equation of Tangents and Normals?
If the point on the curve is (x1, y1) and the slope of the tangent is m, and the slope of the normal is -1/m, then the equation of the tangent is,
y – y1 = m(x – x1)
And, the equation of normal is,
y – y1 = -1/m(x – x1)
How many Tangents are in a Circle?
We can draw infinite tangents to circle. But, we have to note that from a particular point outside the circle, we can draw only two tangents to a circle.
When is the Tangent Parallel to the Y-axis?
If the tangent of any curve at the given point is of the form,
x = h
where h is any constant. Then we say that the tangent is parallel to the y-axis.
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Integrals are the reverse process of differentiation. They are also called anti-derivatives and are used to find the areas and volumes of the arbitrary shapes for which there are no formulas available to us. Indefinite integrals simply calculate the anti-derivative of the function, while the definit
5 min read
Evaluating Definite Integrals
Integration, as the name suggests is used to integrate something. In mathematics, integration is the method used to integrate functions. The other word for integration can be summation as it is used, to sum up, the entire function or in a graphical way, used to find the area under the curve function
9 min read
Properties of Definite Integrals
Properties of Definite Integrals: An integral that has a limit is known as a definite integral. It has an upper limit and a lower limit. It is represented as [Tex]\int_{a}^{b}[/Tex]f(x) = F(b) â F(a) There are many properties regarding definite integral. We will discuss each property one by one with
8 min read
Definite Integrals of Piecewise Functions
Imagine a graph with a function drawn on it, it can be a straight line or a curve or anything as long as it is a function. Now, this is just one function on the graph, can 2 functions simultaneously occur on the graph? Imagine two functions simultaneously occurring on the graph, say, a straight line
8 min read
Improper Integrals
Improper integrals are definite integrals where one or both of the boundaries are at infinity or where the Integrand has a vertical asymptote in the interval of integration. Computing the area up to infinity seems like an intractable problem, but through some clever manipulation, such problems can b
5 min read
Riemann Sums
Riemann Sum is a certain kind of approximation of an integral by a finite sum. A Riemann sum is the sum of rectangles or trapezoids that approximate vertical slices of the area in question. German mathematician Bernhard Riemann developed the concept of Riemann Sums. In this article, we will look int
7 min read
Riemann Sums in Summation Notation
Riemann sums allow us to calculate the area under the curve for any arbitrary function. These formulations help us define the definite integral. The basic idea behind these sums is to divide the area that is supposed to be calculated into small rectangles and calculate the sum of their areas. These
8 min read
Trapezoidal Rule
The Trapezoidal Rule is a fundamental method in numerical integration used to approximate the value of a definite integral of the form bâ«a f(x) dx. It estimates the area under the curve y = f(x) by dividing the interval [a, b] into smaller subintervals and approximating the region under the curve as
13 min read
Definite Integral as the Limit of a Riemann Sum
Definite integrals are an important part of calculus. They are used to calculate the areas, volumes, etc of arbitrary shapes for which formulas are not defined. Analytically they are just indefinite integrals with limits on top of them, but graphically they represent the area under the curve. The li
7 min read
Antiderivative: Integration as Inverse Process of Differentiation
An antiderivative is a function that reverses the process of differentiation. It is also known as the indefinite integral. If F(x) is the antiderivative of f(x), it means that: d/dx[F(x)] = f(x) In other words, F(x) is a function whose derivative is f(x). Antiderivatives include a family of function
6 min read
Indefinite Integrals
Integrals are also known as anti-derivatives as integration is the inverse process of differentiation. Instead of differentiating a function, we are given the derivative of a function and are required to calculate the function from the derivative. This process is called integration or anti-different
6 min read
Particular Solutions to Differential Equations
Indefinite integrals are the reverse of the differentiation process. Given a function f(x) and it's derivative f'(x), they help us in calculating the function f(x) from f'(x). These are used almost everywhere in calculus and are thus called the backbone of the field of calculus. Geometrically speaki
7 min read
Integration by U-substitution
Finding integrals is basically a reverse differentiation process. That is why integrals are also called anti-derivatives. Often the functions are straightforward and standard functions that can be integrated easily. It is easier to solve the combination of these functions using the properties of ind
8 min read
Reverse Chain Rule
Integrals are an important part of the theory of calculus. They are very useful in calculating the areas and volumes for arbitrarily complex functions, which otherwise are very hard to compute and are often bad approximations of the area or the volume enclosed by the function. Integrals are the reve
6 min read
Partial Fraction Expansion
If f(x) is a function that is required to be integrated, f(x) is called the Integrand, and the integration of the function without any limits or boundaries is known as the Indefinite Integration. Indefinite integration has its own formulae to make the process of integration easier. However, sometime
9 min read
Trigonometric Substitution: Method, Formula and Solved Examples
Trigonometric substitution is a process in which the substitution of a trigonometric function into another expression takes place. It is used to evaluate integrals or it is a method for finding antiderivatives of functions that contain square roots of quadratic expressions or rational powers of the
7 min read
Chapter 8: Applications of Integrals
Area under Simple Curves
We know how to calculate the areas of some standard curves like rectangles, squares, trapezium, etc. There are formulas for areas of each of these figures, but in real life, these figures are not always perfect. Sometimes it may happen that we have a figure that looks like a square but is not actual
6 min read
Area Between Two Curves: Formula, Definition and Examples
Area Between Two Curves in Calculus is one of the applications of Integration. It helps us calculate the area bounded between two or more curves using the integration. As we know Integration in calculus is defined as the continuous summation of very small units. The topic "Area Between Two Curves" h
7 min read
Area between Polar Curves
Coordinate systems allow the mathematical formulation of the position and behavior of a body in space. These systems are used almost everywhere in real life. Usually, the rectangular Cartesian coordinate system is seen, but there is another type of coordinate system which is useful for certain kinds
6 min read
Area as Definite Integral
Integrals are an integral part of calculus. They represent summation, for functions which are not as straightforward as standard functions, integrals help us to calculate the sum and their areas and give us the flexibility to work with any type of function we want to work with. The areas for the sta
8 min read
Chapter 9: Differential Equations
Differential Equations
A differential equation is a mathematical equation that relates a function with its derivatives. Differential Equations come into play in a variety of applications such as Physics, Chemistry, Biology, Economics, etc. Differential equations allow us to predict the future behavior of systems by captur
13 min read
Particular Solutions to Differential Equations
Indefinite integrals are the reverse of the differentiation process. Given a function f(x) and it's derivative f'(x), they help us in calculating the function f(x) from f'(x). These are used almost everywhere in calculus and are thus called the backbone of the field of calculus. Geometrically speaki
7 min read
Homogeneous Differential Equations
Homogeneous Differential Equations are differential equations with homogenous functions. They are equations containing a differentiation operator, a function, and a set of variables. The general form of the homogeneous differential equation is f(x, y).dy + g(x, y).dx = 0, where f(x, y) and h(x, y) i
9 min read
Separable Differential Equations
Separable differential equations are a special type of ordinary differential equation (ODE) that can be solved by separating the variables and integrating each side separately. Any differential equation that can be written in form of y' = f(x).g(y), is called a separable differential equation. Basic
8 min read
Exact Equations and Integrating Factors
Differential Equations are used to describe a lot of physical phenomena. They help us to observe something happening in real life and put it in a mathematical form. At this level, we are mostly concerned with linear and first-order differential equations. A differential equation in âyâ is linear if
10 min read
Implicit Differentiation
Implicit Differentiation is the process of differentiation in which we differentiate the implicit function without converting it into an explicit function. For example, we need to find the slope of a circle with an origin at 0 and a radius r. Its equation is given as x2 + y2 = r2. Now, to find the s
6 min read
Implicit differentiation - Advanced Examples
In the previous article, we have discussed the introduction part and some basic examples of Implicit differentiation. So in this article, we will discuss some advanced examples of implicit differentiation. Table of Content Implicit DifferentiationMethod to solveImplicit differentiation Formula Solve
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Advanced Differentiation
Derivatives are used to measure the rate of change of any quantity. This process is called differentiation. It can be considered as a building block of the theory of calculus. Geometrically speaking, the derivative of any function at a particular point gives the slope of the tangent at that point of
8 min read
Disguised Derivatives - Advanced differentiation | Class 12 Maths
The dictionary meaning of âdisguiseâ is âunrecognizableâ. Disguised derivative means âunrecognized derivativeâ. In this type of problem, the definition of derivative is hidden in the form of a limit. At a glance, the problem seems to be solvable using limit properties but it is much easier to solve
6 min read
Derivative of Inverse Trigonometric Functions
Derivative of Inverse Trigonometric Function refers to the rate of change in Inverse Trigonometric Functions. We know that the derivative of a function is the rate of change in a function with respect to the independent variable. Before learning this, one should know the formulas of differentiation
11 min read
Logarithmic Differentiation
Method of finding a function's derivative by first taking the logarithm and then differentiating is called logarithmic differentiation. This method is specially used when the function is type y = f(x)g(x). In this type of problem where y is a composite function, we first need to take a logarithm, ma
8 min read