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

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Question 12:

Find the condition that the curves and intersect orthogonally.

Answer:

Given: two curves and

To find: the condition that these two curves intersect orthogonally

Explanation: Given

Substituting this value of in another curve equation we get

Taking cube root on both sides, we get

Substituting equation (ii) in equation (i), we get

Hence the point of intersection of the two cures is

Now given

Differentiating this with respect to , we get

Now finding the above differentiation value at the point of intersection i.e., at , we get

Also given

Differentiating this with respect to , we get

Now applying the product rule of differentiation, we get

Now finding the above differentiation value at the point of intersection i.e., at , we get

But the two curves intersect orthogonally, if

Now substituting the values from equation (iii) and equation (iv), we get

Taking cube on both sides we get

Hence this is the condition for the given two curves to intersect orthogonally.