If the tangent at (1, 7) to the curve x2 = y – 6 touches the circle x2 + y2 + 16x + 12y + c = 0 then the value of c is

Engineering

Mathematics

Tangent of Parabola

Question

If the tangent at (1, 7) to the curve x² = y – 6 touches the circle x² + y² + 16x + 12y + c = 0 then the value of c is

195

185

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x² = y – 6

2x = y'

y' at (1, 7) = 2

$\frac{k - 7}{h - 1}$ = 2

⇒ k – 7 = 2h – 2

⇒ 2h – k = –5 …….(i)

Now, $\frac{k + 6}{h + 8} = \frac{- 1}{2}$

⇒ 2k + 12 = – h – 8

⇒ h + 2k = –20 ........(ii)

from (i) and (ii)

h = –6 , k = –7

x² + y² + 16x + 12 y + c = 0 ⇒ 36 + 49 – 96 – 84 + c = 0 ⇒ c = 95.