Q. 55.0( 3 Votes )

# Show that the curves 2x = y^{2} and 2xy = k cut at right angles, if k^{2} = 8.

Answer :

Given:

Curves 2x = y^{2} ...(1)

& 2xy = k ...(2)

We have to prove that two curves cut at right angles if k^{2} = 8

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

⇒ 2x = y^{2}

⇒ 2 = 2y.

⇒

m_{1} ...(3)

⇒ 2xy = k

Differentiating above w.r.t x,

⇒ 2(1×) = 0

⇒ = 0

m_{2} ...(4)

Since m_{1} and m_{2} cuts orthogonally,

⇒ ×1

⇒ 1

⇒ x = 1

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

2xy = k & 2x = y^{2}

⇒ (y^{2})y = k

⇒ y^{3} = k

⇒ y

Substituting y in 2x = y^{2},we get,

⇒ 2x = ()^{2}

⇒ 2×1

⇒ 2

⇒ 2^{3}

⇒ 8

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