HOW TO FIND EQUATION OF TANGENT TO THE CURVE

To find the equation of the tangent, we need to have the following things.

(i) A point on the curve on which the tangent line is passing through 

(ii)  Slope of the tangent line.

Note :

We may find the slope of the tangent line by finding the first derivative of the curve.

Equation of Tangent at a Point

Step 1 :

Find the value of dy/dx using first derivative.

Here dy/dx stands for slope of the tangent line at any point. To find the slope of the tangent line at a particular point, we have to apply the given point in the general slope.

Step 2 :

Let us consider the given point as (x₁, y₁)

Step 3 :

By applying the value of slope instead of the variable "m" and applying the values of (x₁ , y₁) in the formula given below, we find the equation of the tangent line.

(y - y₁)  =  m (x - x₁)

Let us look into some example problems to understand the above concept.

Example 1 :

Find the equation of the tangent to the parabola y² = 12x at the point (3, -6).

Solution :

y² = 12x 

Differentiate with respect to "x",

2y (dy/dx)  =  12(1)

m  =  dy/dx  =  12/2y  ==>  6/y

Slope at the point (3, -6)

m  =  6/(-6)  ==> -1

Equation of Tangent :

(y - y₁)  =  m(x - x₁)

(y - (-6)) =  (-1)(x - 3)

y + 6  =  -x + 3

x + y + 6 - 3  =  0 

x + y + 3  =  0

Example 2 :

Find the equation of the tangent to the parabola x² + 2x - 4y + 4 = 0 at the point (0, 1).

Solution :

Equation of the curve is x² + 2x - 4y + 4  =  0

Differentiate with respect to "x",

2x + 2(1) - 4 (dy/dx) + 0

4(dy/dx)  =  2x + 2

dy/dx  =  2(x + 1)/4

=  (x + 1)/2

(dy/dx) _{(0, 1)}  =  (0 + 1)/2  ==>  1/2

Slope  m  =  1/2

Equation of Tangent :

(y - y₁)  =  m(x - x₁)

(y - 1) =  (1/2)(x - 0)

2y - 2  =  x 

x - 2y + 2  =  0

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