Question

If the common tangents to the parabola, $x^2=4y$ and the circle, $x^2+y^2=4$ intersect at the point P, then the distance of P from the origin, is :

• A. $\sqrt{2}+1$
• B. $2(3+2\sqrt{2})$
• C. $2(\sqrt{2}+1)$
• D. $3+2\sqrt{2}$

Answer 1

Answer: (C)

tangent to $x^2+y^2=4$
$y=mx\pm 2\sqrt{1+m^2}$
$x^2=4y$
$x^2=4mx+8\sqrt{1+m^2}$
$x^2=4mx-8\sqrt{1+m^2}=0$
$D=0$
$16m^2+4.8\sqrt{1+m^2}=0$
$m^2+2\sqrt{1+m^2}=0$
or $m^2=\sqrt{1+m^2}<br/>m^4=4+4m^2$
$m^4-4m^2-4=0$
$m^2=\frac{4\pm \sqrt{16+16}}{2}$
$=\frac{4\pm 4\sqrt{2}}{2}$
$m^2=2+2\sqrt{2}$