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If the chord of contact of tangents from a point P to the parabola $\mathrm{y^2=4 a x}$ touches the parabola $\mathrm{x^2=4 b y}$ , then the locus of P is
Option: 1
a pair of straight lines

Option: 2
a circle

Option: 3
a hyperbola

Option: 4
a parabola

Answers (1)

best_answer

Let point P be $\mathrm{(h, k).}$

The chord of contact of $\mathrm{P(h, k)}$ w.r.t. $\mathrm{y^2=4 a x}$ is

$\mathrm{ T \equiv k y-2 a(x+h)=0 }$

Or $\mathrm{\quad x=\frac{k}{2 a} y-h}$ $\mathrm{..........(1)}$

Any tangent to $\mathrm{ x^2=4 b y}$ is

$\mathrm{ x=m y+\frac{b}{m} }$ $.........(2)$
On comparing (1) and (2), we obtain

$\mathrm{ m=\frac{k}{2 a} \text { and }-h=\frac{b}{m} }$
On eliminating m,

$\mathrm{ -\frac{b}{h}=\frac{k}{2 a} }$

$\therefore \quad$ the locus of $\mathrm{(h, k)}$ is

$\mathrm{ x y=-2 a b }$
which is the equation of rectangular hyperbola.

Posted by

Ajit Kumar Dubey

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