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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 circle
 


Option: 2

 a parabola
 


Option: 3

a straight line
 


Option: 4

 none of these


Answers (1)

best_answer

Let \mathrm{P \equiv\left(x_1, y_1\right)}

Chord of contact \mathrm{ y y_1=2 a\left(x+x_1\right) \Rightarrow x=\frac{y_1}{2 a} y-x_1 \ldots (i)}

Tangent to \mathrm{ x^2=4 b y ~is~ x=m y+\frac{b}{m}\ldots (ii)}

Compare (i) and (ii)  \mathrm{ m=\frac{y_1}{2 a} ~and ~x_1=-\frac{b}{m}, \quad x_1=-\frac{b}{\frac{y_1}{2 a}} \Rightarrow x_1 y_1=-2 a b}Required locus \mathrm{ x y=-2 a b}  which is hyperbola.

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Ritika Harsh

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