Question

The tangent at (1, 7) to the curves $x^2=y-6$ 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 $95$

Curve $x^2=y-6$

$\Rightarrow 2x=\frac{dy}{dx}-0$

$\Rightarrow {\left[\frac{dy}{dx}\right]}_{(1,7)}=2$

Equation of tangent at $(1,7)$

$y-7=2(x-1)$

$\Rightarrow 2x-y+5=0$ ..........$\left(1\right)$

Line $\left(1\right)$ touches the circle $x^2+y^2+16x+12y+c=0$

$\therefore p=r$

$\left|\frac{2\times -8-(-6)+5}{\sqrt{4+1}}\right|=\sqrt{64+36-c}$

$\Rightarrow \frac{25}{5}=100-c$

$\Rightarrow 5=100-c$

$\therefore c=100-5=95$