The tangent at (1,7) to the parabola x2=y−6 is x=21(y+7)−6 ⇒2x=y+7−12 ⇒y=2x+5
which is also a tangent to the circle x2+y2+16x+12y+c=0 ∴x2+(2x+5)2+16x+12(2x+5)+c=0 ⇒5x2+60x+85+c=0
must have equal roots.
Let α and β be the roots of the equation.
Then, α+β=−12⇒α=−6(∵α=β) ∴x=−6 and y=2x+5=−7 ⇒ Point of contact is (−6,−7)