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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