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  2. Prove that the curves x = y ^ 2 and x y = k cut at right angles* if 8 k ^ 2 = 1.
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23) Prove that the curves x = y^2 and xy = k cut at right angles* if \: \: 8k ^ 2 = 1.

Answers (1)
G Gautam harsolia

Let suppose, Curve x = y^2 and xy = k cut at the right  angle
then the slope of their tangent also cut  at the right angle
means,
\left ( \frac{dy}{dx} \right )_a \times \left ( \frac{dy}{dx} \right )_b = -1                                                                   -(i)
2y\left ( \frac{dy}{dx} \right )_a = 1 \Rightarrow \left ( \frac{dy}{dx} \right )_a = \frac{1}{2y}
\left ( \frac{dy}{dx} \right )_b = \frac{-k}{x^2}
Now these values in equation (i)
\frac{1}{2y} \times \frac{-k}{x^2} = -1\\ -k = -2yx^2\\ k =2(xy)(x)\\ k = 2k(k^{\frac{2}{3}}) \ \ \ \ \left ( x = y^2 \Rightarrow y^2y = k \Rightarrow y = k^{\frac{1}{3}} \ and \ x = k^{\frac{2}{3}} \right ) \\ 2(k^{\frac{2}{3}}) = 1\\ \left ( 2(k^{\frac{2}{3}}) \right )^3 = 1^3\\ 8k^2 = 1
                              Hence proved

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