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Let a, r, s, t be non-zero real numbers....

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 is

A

`=(1)/(t)`

B

`(t^(2) + 1)/(t)`

C

`(1)/(t)`

D

`(t^(2) - 1)/(t)`

Text Solution

Verified by Experts

The correct Answer is:
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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Knowledge Check

  • 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 point on the parabola y^(2) = 4ax . Suppose the PQ si the focal chord and line QR and PK are parallel, where K is point (2a, 0) It st = 1, then the tangent at P and the normal at S to the parabola meet at a point whose ordinate is

    A
    `((t^(2) + 1)^(2))/(2t^(3))`
    B
    `(a(t^(2) + 1)^(2))/(2t^(3))`
    C
    `(a(t^(2) + 1)^(2))/(t^(3))`
    D
    `(a(t^(2) + 2)^(2))/(t^(3))`

  • If P(at_(1)^(2), 2at_(1))" and Q(at_(2)^(2), 2at_(2)) are two points on the parabola at y^(2)=4ax , then that area of the triangle formed by the tangents at P and Q and the chord PQ, is

    A
    `1/2a^(2)|t_(1)-t_(2)|^(3)`
    B
    `1/2a^(2)|t_(1)-t_(2)|^(2)`
    C
    `a^(2)|t_(1)-t_(2)|^(3)`
    D
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    B
    algebraic sum of the ordinates of the points P,Q and R vanishes
    C
    centeroid of the traingle PQR lies on the axis of the parabola
    D
    Circle cicrcumscribing the traingle PQR passes through the vertex of the parabola.

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