VITEEE 2012 Paper

The tangent at (1, 7) to the curve x2 = y - 6 is
x=

1

2

(y + 7) - 6
⇒ 2x = y + 7-12
⇒ y = 2x + 5
which is also tangent to the circle
x2+y2 + 16x + 12y + c = 0
i.e., x2 + (2x+5)2 + 16x + 12 (2x + 5) + c = 0
5x2 + 60x + 85 + c = 0, which must have equal roots.
Let α and β are the roots of the equation.
Then α + β = -12 ⇒ α = - 6 (Since α = β)
∴ x = - 6 ,y = 2x + 5 = -7
∴ Point of contact is (-6, -7).