Q. 125.0( 2 Votes )

Find the condition that the curves 2x = y2 and 2xy = k intersect orthogonally.

Answer :

Given: two curves 2x = y2 and 2xy = k


To find: the condition that these two curves intersect orthogonally


Explanation: Given 2xy = k



Substituting this value of y in another curve equation i.e., 2x = y2, 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 2x = y2


Differentiating this with respect to x, we get





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



Also given 2xy = k


Differentiating this with respect to x, 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


m1.m2 = -1


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








Taking cube on both sides we get


k2 = 23 = 8


k = 2√2


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


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