# NCERT Class 12-Mathematics: Chapter –6 Application of Derivatives Part 14

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Question 16:

Prove that the curves and touch each other at the point .

We have,

Since, both the curves touch each other at i.e. curves are passing through

And

And

Thus, we see that slope of both the curves are equal to each other i.e., at the point

Hence, both the curves touch each other.

Question 17:

Find the equation of the normal lines to the curve which are parallel to the line .

Given: equation of the curve , equation of line

To find: the equation of the normal lines to the given curve which are parallel to the given line

Explanation:

Now given equation of curve as

Differentiating this with respect to , we get

Now let slope of the normal to the curve be is given by

Substituting value from equation (i), we get

The given equation of the line is

the slope of this line is

Since, slope of normal to the curve should be equal to the slope of the line which is parallel to the curve,

Substituting values from equation (ii) and (iii), we get

Now substituting in the equation of the curve, we get

But from equation (iv)

Thus the points at which normal to the given curve is parallel to the given line are and

Now the equations of the normal are given by

and

Hence the equation of the normal lines to the curve which are parallel to the line are .