Question

Let tangents be drawn from the point $P(1,3)$ on the ellipse $2x^2+y^2=1$ and points of contact be $A$ and B. The line AB cuts the hyperbola $2x^2-y^2=1$ at $M\&N$ and intersects the parabola $y^2=16x$ at $Q$ and $R$.
If the line $AB$ is tangent to a circle with centre $(4,2)$ and the circle intersects $x^2+y^2-5x-2y$ $+c=0$ 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 $2x+3y-1=0$
$=\left|\frac{2.4+3.2-1}{\sqrt{2^2+3^2}}\right|=\sqrt{13}$
$\therefore$ Equation of circle will be
$(x-4{)}^2+(y-2{)}^2=(\sqrt{13}{)}^2$
$\Rightarrow x^2+y^2-8x-4y+7=0$ ....(iii)
$\Theta$ circle (iii) is orthogonal to $x^2+y^2-5x-2y+c=0$
$\therefore 2(-4)\left(\frac{-5}{2}\right)+2(-2)(-1)=7+c$
$\Rightarrow 24=7+c\Rightarrow c=17$