# HOW TO FIND EQUATION OF TANGENT TO THE CURVE WITHOUT DERIVATIVE

## About the topic "How to find equation of tangent to the curve without derivative"

How to find equation of tangent to the curve without derivative :

Here we are going to see how to find equation of tangent to the curve without derivative

To find the equation of the tangent, we need to do the following changes in the given equation of curve.

Replace

x2  as xx1,

y2 as yy1,

x  as (x + x1)/2

y as (y + y1)/2

xy as (xy1 + yx1)/2

Now we have to apply the given point instead of (x1, y1) in the equation.

Let us look into some examples to understand the above concept.

Example 1 :

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

Solution :

In the first step, we are going to replace y2 = yy1 and x = (x + x1)/2

Equation of tangent at (x1, y1)

yy1 = 12(x + x1)/2

y(6)  =  6(x + 3)

6y  =  6x + 18

6x - 6y + 18 = 0

Dividing the equation by 6, we get

x - y + 3  =  0

Hence equation of tangent is x - y + 3  =  0

Example 2 :

Find the equation of the tangent to the parabola x2 + x − 2y + 2 = 0 at (1, 2)

Solution :

In the first step, we are going to replace x2 = xx1 and x = (x + x1)/2 and y = (y + y1)/2

Equation of tangent at (x1, y1)

xx1 + ((x + x1)/2) − 2(y + y1)/2 + 2 = 0

x(1) + ((x + 1)/2) − (y + 2) + 2 = 0

2x + x + 1 - 2(y + 2) + 4 = 0

3x - 2y - 4 + 1 + 4 = 0

3x - 2y + 1 = 0

Hence equation of tangent is 3x - 2y + 1 = 0

Example 3 :

Find the equation of the tangent to the hyperbola 9x2- 5y2 = 31 at (2, -1)

Solution :

In the first step, we are going to replace x2 = xx1 and y2 = yy1

Equation of tangent at (x1, y1)

9xx1- 5yy= 31

Equation of tangent at (2, -1)

9x(2) - 5y(-1) = 31

18x + 5y  =  31

Hence equation of tangent is 18x + 5y  =  31

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