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

• Correct answer
  A

  $95$
• B

  $195$
• C

  $185$
• D

  $85$

Solution

Differentiating $x^2=y-6$ with respect to $x,$

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

${\left.\frac{dy}{dx}\right|}_{\left(\mathrm{1,7}\right)}=2\times 1=2$

Hence, equation of tangent is $y-7=2(x-1)$

$\Rightarrow y=2x+5$

Distance from centre =

$\left[\frac{-6-2\left(-8\right)-5}{\sqrt{1^2+2^2}}\right]$

$\Rightarrow \left|\frac{5}{\sqrt{5}}\right|=\sqrt{5}$

Radius of circle $= \sqrt{8^2+6^2-c}$

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

$\Rightarrow c=95$

Hence, the correct answer is option: 'A'.