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If the tangent at (1,7) to curve x^(2)=y...

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

A

(6,7)

B

(-6, 7)

C

(6, -7)

D

(-6,-7)

Text Solution

Verified by Experts

The correct Answer is:
D
The tangent at (1, 7) to the parabola `x^(2) = y - 6x` is
`x (1) = (1)/(2) (y + 7) - 6`
[replacing `x^(2) to xx_(1)` and `2y to y + y_(1)`]
`implies 2x = y + 7 - 12`
`implies y = 2x + 5`
Which is also tangents to the circle
`x^(2) + y^(2) + 16x + 12 y + c = 0`
i.e., `x^(2) + (2x + 5)^(2) + 16x + 12 (2x + 5) + C = 0` must have equal, rools i.e., `alpha = beta`
`implies 5x^(2) + 60x + 85 + c = 0`
`implies alpha + beta = (-60)/(5)`
`implies alpha = - 6`
`:. x = - 6` and `y = 2x + 5 = - 7`
`:.` Point of contact is (-6, -7).
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