Home
Class 12
MATHS
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)`
Promotional Banner

Topper's Solved these Questions

  • PARABOLA

    IIT JEE PREVIOUS YEAR|Exercise TOPIC 2 EQUATION OF TANGENTS AND PROPERTIES (FILL IN THE BLANK )|1 Videos

  • PARABOLA

    IIT JEE PREVIOUS YEAR|Exercise TOPIC 2 EQUATION OF TANGENTS AND PROPERTIES (ANALYTICAL & DESCRIPTIVE )|2 Videos

  • PARABOLA

    IIT JEE PREVIOUS YEAR|Exercise TOPIC 2 EQUATION OF TANGENTS AND PROPERTIES OBJECTIVES QUESTIONS II (ONE OR MORE THAN CORRECT OPTION )|1 Videos

  • MISCELLANEOUS

    IIT JEE PREVIOUS YEAR|Exercise MISCELLANEOUS|87 Videos

  • PERMUTATIONS AND COMBINATIONS

    IIT JEE PREVIOUS YEAR|Exercise Dearrangement and Number of Divisors (Fill in the Blank )|1 Videos

Similar Questions

Explore conceptually related problems

Let a, r, s, t be non-zero real numbers. Let P(at^(2),2at),Q(ar^(2),2ar)andS(as^(2),2as) be distinct points on the parabola y^(2)=4ax . Suppose that PQ is the focal chord and lines QR and PK are parallel, where K the point (2a,0). If st=1, then the tangent at P and the normal at S to the parabola meet at a point whose ordinate is

If tangent at P and Q to the parabola y^(2)=4ax intersect at R then prove that mid point the parabola,where M is the mid point of P and Q.

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

The point P on the parabola y^(2)=4ax for which | PR-PQ | is maximum, where R(-a, 0) and Q (0, a) are two points,

Let P and Q be the points on the parabola y^(2)=4x so that the line segment PQ subtends right angle If PQ intersects the axis of the parabola at R, then the distance of the vertex from R is

If the tangents at the points P and Q on the parabola y^2 = 4ax meet at R and S is its focus, prove that SR^2 = SP.SQ .

Let P(at_(1)^(2),2at_(1)) and Q(at_(2)^(@),2at_(2)) are two points on the parabola y^(2)=4ax Then,the equation of chord is y(t_(1)+t_(2))=2x+2at_(1)t_(2)

Let P(2, 2) and Q(1//2, -1) be two points on the parabola y^(2)=2x , The coordinates of the point R on the parabola y^(2) =2x where the tangent to the curve is parallel to the chord PQ, are