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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 a/an
 

Option: 1

 circle
 


Option: 2

 parabola
 


Option: 3

 ellipse
 


Option: 4

 hyperbola
 


Answers (1)

best_answer

 Let \mathrm{P(h, k)}  be a point. Then the chord of contact of tangents from P to \mathrm{\mathrm{y}^2=4 a x ~is~ k y=2 a(x+h) \ldots(1)}

This touches the parabola \mathrm{x^2=4 b y}. So, it should be of the form

\mathrm{x=m y+\frac{b}{m} }
Equation (i) can be re-written as

\mathrm{x=\frac{k}{2 a} y-h }
Since (ii) and (iii) represent the same line.
\mathrm{\therefore \quad m=\frac{k}{2 a} ~and ~\frac{b}{m}=-h}

Eliminating \mathrm{m} from these two equations, we get\mathrm{2 a b=-h k}

Hence, the locus of \mathrm{P(h, k)}  is \mathrm{x y=-2 a b,} which is a hyperbola.

Posted by

Pankaj Sanodiya

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