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Let a, r, s, t be non-zero real numbers. Let `P(at^2, 2at), Q, R(ar^2, 2ar) and S(as^2, 2as)` be distinct points onthe parabola `y^2 = 4ax`. Suppose that PQ is the focal chord and lines QR and PK are parallel, where K isthe point (2a, 0). The value of r isA. `=(1)/(t)`B. `(t^(2) + 1)/(t)`C. `(1)/(t)`D. `(t^(2) - 1)/(t)` |
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Answer» Correct Answer - D Plan (I) If `P(at^(2), 1at)` is one end point of focal chord of parabola `y^(2) = 4ax`, then other end point is `((a)/(t^(2)), (2a)/(t))` (ii) Slope of line joining two points `(x_(1), y_(1))` and `(x_(2), y_(2))` is given by `(y_(2) - y_(1))/(x_(2) - x_(1))` If PQ is focal chord, then coordinates of Q will be `((a)/(t^(2)),(2a)/(t))` Now, slope of QR = slope of PK `(2ar + (2a)/(t))/(ar^(2) - (a)/(t^(2))) = (2 at)/(at^(2) - 2a) implies (r + 1//t)/(r^(2) - 1//t^(2)) = (t)/(t^(2) - 2)` `implies (1)/(r - (1)/(t)) = (t)/(t^(2) - 2) implies r - (1)/(t) = (t^(2) - 2)/(t) = t - (2)/(t)` `implies r = t - (1)/(t) = (t^(2) - 1)/(t)` |
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