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

Answer 1

The correct option is B $95$

Given equation of curve is $x^2=y-6$

$\Rightarrow y=x^2+6$

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

Slope of tangent is $m={\left|\frac{dy}{dx}\right|}_{(1,7)}=2\times 1=2$

Equation of tangent at the point $(1,7)$ is given by

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

$\Rightarrow y-7=2x-2$

$\Rightarrow 2x-y+5=0$

Given equation of circle is

$x^2+y^2+16x+12y+c=0$

$\Rightarrow (x+8{)}^2+(y+6{)}^2+c-64-36=0$

$\Rightarrow (x+8{)}^2+(y+6{)}^2=100-c$

If the tangent touches the circle, then

Distance from centre $=$ radius

$\Rightarrow$ Distance of $(-8,-6)$ from $2x-y+5=0$ is the radius

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

$\Rightarrow Radius=\sqrt{100-c}=\sqrt{5}$

$\Rightarrow c=95$