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The normal to the ellipse 3x² + 4y² = 12 at a point P is parallel to the line 2x + y = 4, and the tangent at P passes through Q(4, 4). Find the length PQ.
- 5*sqrt(5)/2
- sqrt(221)/2
- sqrt(157)/2
- sqrt(61)/2
Correct answer: 5*sqrt(5)/2
Solution
The normal-slope condition gives tan t = -sqrt(3), and the tangent through Q(4,4) selects P = (-1, 3/2); then PQ = sqrt(5² + (5/2)²) = 5*sqrt(5)/2.
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