Latus rectum of the conic satisfying the differential equation $x d y + y d x = 0$ and passing through the point $(2, 8)$ is :

4 \sqrt{2}

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8

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8 \sqrt{2}

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16

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Solution

The correct option is C $8 \sqrt{2}$
The differential equation is
$x d y + y d x = 0$
$d (x y) = 0$
By integrating we get,
$x y = k$ ..........(1)

Equation (1) passes through the point $(2, 8)$ , so
$(2) (8) = k$
$k = 16$

So, equation of the conic is
$x y = 16$
which is a rectangular hyperbola $(x y = c^{2})$ , where $c = 4$

Length of latus rectum for rectangular hyperbola is $2 \sqrt{2} c = 8 \sqrt{2}$

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