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A point P is such that the line through P perpendicular to the chord of contact of tangents drawn from P to the parabola y² = 16x is itself a tangent to the parabola x² = 12y. Find the locus of P.
- 4x + 3y + 64 = 0
- 4x + 3y + 16 = 0
- 8x + 3y + 16 = 0
- 8x + 3y + 64 = 0
Correct answer: 8x + 3y + 64 = 0
Solution
The chord of contact from P=(h,k) to y²=16x is ky = 8(x+h), slope 8/k. The perpendicular through P has slope -k/8 and equation kx + 8y = k(h+8). Setting the discriminant to zero for tangency with x²=12y gives the locus 8x + 3y + 64 = 0.
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