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. 195
• B. 185
• C. 85
• D. 95

Answer 1

Answer: (D)

95

Answer 2

Equation of tangent at $(1,7)$ to curve $x^{2}=y-6$ is
$x-1=\frac{1}{2}(y+7)-6$
$2 x-y+5=0\,\,\,\,\,\,\,...(i)$
Centre of circle $=(-8,-6)$
Radius of circle $=\sqrt{64+36-c}=\sqrt{100-c}$
$\because$ Line (i) touches the circle
$\therefore \left|\frac{2(-8)-(-6)+5}{\sqrt{4+1}}\right|=\sqrt{100-c}$
$\sqrt{5}=\sqrt{100-c}$
$\Rightarrow c =95$