Show that the curves 4x = y2 and 4xy = k cut at right angles, if k2 = 512.

Given:


Curves 4x = y2 ...(1)


& 4xy = k ...(2)


We have to prove that two curves cut at right angles if k2 = 512


Now ,Differentiating curves (1) & (2) w.r.t x, we get


4x = y2


4 = 2y.



m1 ...(3)


4xy = k


Differentiating above w.r.t x,


4(1×) = 0


= 0



m2 ...(4)



Since m1 and m2 cuts orthogonally,


×1


1


x = 2


Now , Solving (1) & (2),we get,


4xy = k & 4x = y2


(y2)y = k


y3 = k


y


Substituting y in 4x = y2,we get,


4x = ()2


4×2


8


83


512


4