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For the hyperbola x^2/16 - y^2/9 = 1, the tangent at a variable point…
Question
For the hyperbola x^2/16 - y^2/9 = 1, the tangent at a variable point P meets the two asymptotes 3x - 4y = 0 and 3x + 4y = 0 at the points A and B. For a particular position of P the chord AB has length 10. For that position, find the value of OA^2 + OB^2, where O is the origin.
✓ Verified answer: 114checked by our engine — not a guess
Step-by-step solution
Parametrise P = (4 sec t, 3 tan t). The tangent at P is x sec t /4 - y tan t /3 = 1.
Intersect with asymptote y = (3/4)x: substitute to get A; intersect with y = -(3/4)x to get B.
Carrying out the algebra (or by direct symbolic computation) one finds the two clean identities, valid for all t:
AB^2 = 100 sec^2 t - 64, OA^2 + OB^2 = 100 sec^2 t - 50.
Subtracting, OA^2 + OB^2 - AB^2 = 14, a constant.
Given AB = 10, AB^2 = 100, so OA^2 + OB^2 = 100 + 14 = 114.
(Consistency: AB^2 = 100 gives 100 sec^2 t = 164, a valid sec^2 t = 1.64 > 1.)
Final answer114
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