Find the length of sub normal to the curve $x^{2} + y^{2} + x y = 7$ at the point $(1, - 3)$ .

15

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20

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25

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39

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Solution

The correct option is B $15$
$x^{2} + y^{2} + x y = 7$
Differentiating w.r.t $x$
$2 x + 2 y \frac{d y}{d x} + y + x \frac{d y}{d x} = 0$
$\Rightarrow \frac{d y}{d x} = - \frac{2 x + y}{2 y + x}$
$∴ {(\frac{d y}{d x})}_{(1, - 3)} = - 5$
Hence length of sub normal is, $= {(y . \frac{d y}{d x})}_{(1, - 3)} = 15$

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