If the tangent at (1, 7) to the curve x ^ 2 = y - 6 touch the...

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

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

{\left\dfrac{dy}{dx}\right|}_{\left(1,7\right)} = 2 \times 1 = 2

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

$⇒ y = 2x + 5$

Distance from centre =

\left[\dfrac{- 6 - 2\left(- 8\right) - 5}{\sqrt{{1}^{2} + {2}^{2}}}\right]$<br /><br />$⇒ \left|\dfrac{5}{\sqrt{5}}\right| = \sqrt{5}$<br /><br />Radius of circle$= \sqrt{{8}^{2} + {6}^{2} - c}$<br /><br />$⇒ \sqrt{100 - c} = \sqrt{5}$<br /><br />$⇒ c = 95