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MHT CET202519 Apr 2025Evening ShiftMathematicsApplication of DerivativesActual

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

Options

  1. A85
  2. B95
  3. C185
  4. D195

Correct answer

B. 95

Step-by-step solution

Finding the tangent equation: For the curve x^2 = y - 6 , differentiate to find the slope: 2x = dy dx , so at (1,7) , m = 2(1) = 2 . The tangent line is y - 7 = 2(x - 1) , which simplifies to 2x - y + 5 = 0 . Circle analysis: The circle x^2 + y^2 + 16x + 12y + C = 0 has center (-8, -6) and radius R = 100 - C . Tangency condition: The distance from the center (-8, -6) to the line 2x - y + 5 = 0 is d = |2(-8) - 1(-6) + 5| 2^2 + (-1)^2 = 5 5 = 5 . Set d = R , so 5 = 100 - C . Squaring both sides yields 5 = 100 - C , h

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