If the tangent at (1,7) to the curve x²=y touches the circle x²+y²+16x+12y+c=0, then the value of c is

Given equation of curve is x²=y ⇒ y=x²+6 ⇒ dy/dx=2x
Slope of tangent is m = |dy/dx|(1,7) = 2×1 = 2
Equation of tangent at the point (1,7) is given by (y-7)=2(x-1) ⇒ y-7=2x-2 ⇒ 2x-y+5=0
Given equation of circle is x²+y²+16x+12y+c=0 ⇒ (x+8)²+(y+6)²+c-64-36=0 ⇒ (x+8)²+(y+6)²=100-c
If the tangent touches the circle, then
Distance from centre=radius
⇒Distance of (-8,-6) from 2x-y+5=0 is the radius
d=|2(-8)-(-6)+5|/√4+1 = |-16+6+5|/√5 = |-5|/√5 = √5
⇒Radius = √100-c = √5
⇒100-c=5
⇒c=95