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Standard XII

Mathematics

Rectangular Hyperbola

P M and P N a...

Question

$P M$ and $P N$ are the perpendiculars from any point $P$ on the rectangular hyperbola $x y = 8$ to the asymptotes. If the locus of the mid point of MN is a conic, then the least distance of $(1, 1)$ to director circle of the conic is

\sqrt{3}

unit

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\sqrt{2}

unit

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2 \sqrt{3}

unit

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2 \sqrt{5}

unit

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Solution

The correct option is B $\sqrt{2}$ unit

Here, $O M P N$ is rectangle.
$P \equiv (2 \sqrt{2} t, \frac{2 \sqrt{2}}{t})$
Mid point of $M N = O P$
$∴ (h, k) = (\sqrt{2} t, \frac{\sqrt{2}}{t})$
Thus $h \times k = \sqrt{2} t \cdot \frac{\sqrt{2}}{t} = 2$
Thus locus is the rectangular hyperbola $x y = 2$
The director circle to this rectangular hyperbola is the point circle with center $(0, 0)$ .
Thus distance between $(0, 0)$ and $(1, 1)$ is $\sqrt{2}$ unit

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PM and PN are the perpendiculars from any point P on the rectangular hyperbola xy=8 to the asymptotes. If the locus of the mid point of MN is a conic, then the least distance of (1,1) to director circle of the conic is

$P M$ and $P N$ are the perpendiculars from any point $P$ on the rectangular hyperbola $x y = 8$ to the asymptotes. If the locus of the mid point of MN is a conic, then the least distance of $(1, 1)$ to director circle of the conic is