If the tangent at (1,7) to curve `x^(2)=y-6` touches the circle `x^(2)+y^(2)+16x+12y+c=0` then the value of c is

Options

A. 95
B. 195
C. 185
D. 85

Answer

A

Explanation

1 Differentiating curve `x^(2)=y-6` w.r.t. x, we get
`(dy)/(dx)=2x`
Therefore, slope of tangent to the curve at point P(1,7) is
`((dy)/(dx))_((1","7))=2`
Equation of tangent to the curve at point P is
y-7=2(x-1)
`or2x-y+5=0` (1)
Given circle is `x^(2)+y^(2)+16x+12y+c=0`
Center of the circle is C(-8,-6).
Radius of the circle, `r=sqrt(46+36-c)`
Line (!) touches the circle at point M.
So, CM = r
`(|-16+65|)/(sqrt(2^(2)+(-1)^(2)))=sqrt(64+36-c)`
`:." "sqrt(100-c)=sqrt(5)`
`:." "c=95`