Show that the curves 2x = y2 and 2xy = k cut at right angles, if k2 = 8.

Given:

Curves 2x = y2 ...(1)

& 2xy = k ...(2)

We have to prove that two curves cut at right angles if k2 = 8

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

2x = y2

2 = 2y.

m1 ...(3)

2xy = k

Differentiating above w.r.t x,

2(1×) = 0

= 0

m2 ...(4)

Since m1 and m2 cuts orthogonally,

×1

1

x = 1

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

2xy = k & 2x = y2

(y2)y = k

y3 = k

y

Substituting y in 2x = y2,we get,

2x = ()2

2×1

2

23

8

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