Question

The condition that two tangents to the parabola $y^2=4gx$ become normals to the circle $x^2+y^2-2gx-2fy+c=0$ is given by

• A. $g^2>4f^2$
• B. $c^2>2g^2$
• C. $g^2>2f^2$
• D. $f^2>4g^2$

Answer 1

The correct option is D $f^2>4g^2$

Tangent of the given parabola with slope $m$ is $y=mx+\frac{g}{m}$

For it to be a normal to the given circle , it has to pass through the centre of it.
Hence $(g,f)$ has to satisfy tangent equation of parabola.

$\Rightarrow f=mg+\frac{g}{m}$

$m^{2}g-mf+g=0$

Two tangents $\Rightarrow$ the equation has two distinct real roots

which is possible when $f^2-4g^2>0$

$\Rightarrow f^2>4g^2$

Hence, option 'D' is correct.