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
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195
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185
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85
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95
Curve x^2 = y - 6 => y = x^2 + 6. Derivative dy/dx = 2x. At (1, 7), slope m = 2. Tangent: y - 7 = 2(x - 1) => y = 2x + 5 => 2x - y + 5 = 0. Circle x^2 + y^2 + 16x + 12y + c = 0 has center (-8, -6) and radius sqrt(64 + 36 - c) = sqrt(100 - c). Distance from center to tangent = |2(-8) - (-6) + 5| / sqrt(2^2 + (-1)^2) = |-16 + 6 + 5| / sqrt(5) = 5 / sqrt(5) = sqrt(5). Radius^2 = 5 = 100 - c => c = 95.
Differentiating x2 = y - 6 gives the slope of the tangent as 2x, which is 2 at the point (1, 7). The equation of this tangent line is y minus 7 equals 2 times the quantity (x minus 1), which simplifies to 2x minus y plus 5 equals 0. The center of the circle x2 + y2 + 16x + 12y + c = 0 is (-8, -6), and the distance from the center to the tangent line is the absolute value of (2 multiplied by -8 minus -6 plus 5) divided by the square root of (4 plus 1), yielding 5. Setting this radius equal to 5 gives 64 plus 36 plus c equals 25, which solves for c as 95.