Question

Let tangents be drawn from the point $P (1,3)$ on the ellipse $2 x ^2+ y ^2=1$ and points of contact be $A$ and B. The line AB cuts the hyperbola $2 x^2-y^2=1$ at $M \& N$ and intersects the parabola $y^2=16 x$ at $Q$ and $R$.
If the line $AB$ is tangent to a circle with centre $(4,2)$ and the circle intersects $x ^2+ y ^2-5 x -2 y$ $+ c =0$ orthogonally then the value of $c$ is

• A. 5
• B. 11
• C. 17
• D. 23

Answer 1

Answer: (C)

$\Theta$ Radius of circle $=$ perpendicular distance of $(4,2)$ from $2 x+3 y-1=0$
$=\left|\frac{2.4+3.2-1}{\sqrt{2^2+3^2}}\right|=\sqrt{13}$
$\therefore$ Equation of circle will be
$ ( x -4)^2+( y -2)^2=(\sqrt{13})^2 $
$\Rightarrow x ^2+ y ^2-8 x -4 y +7=0$ ....(iii)
$\Theta$ circle (iii) is orthogonal to $x^2+y^2-5 x-2 y+c=0$
$\therefore 2(-4)\left(\frac{-5}{2}\right)+2(-2)(-1)=7+c$
$\Rightarrow 24=7+ c \Rightarrow c =17$