Question

If the tangent at $(1,7)$ to the curve $x^2=y-6$ touches the circle $x^2+y^2+16x+12y+c=0,$ then the value of $c$ is

• A. $95$
• B. $195$
• C. $185$
• D. $85$

Answer 1

The correct option is A $95$

Equation of tangent at $(1,7)$ to $x^2=y-6$ is
$x-\frac{y+7}{2}+6=0$
$\Rightarrow 2x-y+5=0$
Which is tangent to the circle $x^2+y^2+16x+12y+c=0$
$\therefore$ Radius $=$ Perpendicular distance from the centre to tangent
$\Rightarrow \sqrt{8^2+6^2-c}=\left|\frac{-16+6+5}{\sqrt{5}}\right|$
$\Rightarrow \sqrt{8^2+6^2-c}=\sqrt{5}$
$\Rightarrow 8^2+6^2-c=5$
$\therefore c=95$