Question

If the chord of contact of tangent from a point P to the parabola $y^2=a^2$ touches the parabola $x^2+(y-a{)}^2$, then the locus of P is

• A. Straight line
• B. Circle
• C. Parabola
• D. Hyperbola

Answer 1

The correct option is C Parabola

$\begin{array}{c}Letp\left(h,k\right)beapointfromwhere\tan gentsaredrawn\\ tothecircle{x}^2+y^2=a^2\\ equationaschordatcontactis\\ xh+yk=a^2---\left(1\right)\\ Lines\left(1\right)touches{x}^2+{\left(y-a\right)}^2=h^{2}so,\\ \frac{0+ak-a^2}{\sqrt{a^2+k^2}}=a\end{array}$

or, $(k-a{)}^2=h^2+k^2$

or, $h^2=a^2-2ak$.

so it is the equation of a parabola.