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  • If tangents to the parabola   intersect the hyperbola <img alt="\mathrm{\

If tangents to the parabola \mathrm{y^2=4 a x}  intersect the hyperbola \mathrm{\frac{x^2}{a^2}-\frac{y^2}{b^2}=1} at A and B, then find the locus of point of intersection of tangents at A and B.

Option: 1

a^3 y^2+b^4 x=0


Option: 2

a^4 y^2+b^3 x=0


Option: 3

a^3 x^2+b^4 y=0


Option: 4

a^3 y^2-b^4 x=0


Answers (1)

best_answer

Tangents to parabola intersect the hyperbola at A and B.

Let the point of intersection of tangents at A and B be P(h, k).

So, AB will be chord of contact of hyperbola w.r.t point P.

Thus, equation of AB is

\frac{h x}{a^2}-\frac{k y}{b^2}=1

or \frac{k y}{b^2}=\frac{h x}{a^2}-1$ or $y=\left(\frac{b^2 h}{k a^2}\right) x-\frac{b^2}{k}

This line touches the parabola

So, -\frac{b^2}{k}=\frac{a}{\frac{b^2 h}{k a^2}}

\text { (as } y=m x+c \text { touches the parabola } y^2=4 a x \text { if } c=a / m \text { ) }

\Rightarrow-\frac{b^2}{k}=\frac{k a^3}{b^2 h}

Hence, required locus is y^2=-\frac{b^4}{a^3} x.

Posted by

Irshad Anwar

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