Home
Class 12
MATHS
Let P and Q be distinct points on the pa...

Let P and Q be distinct points on the parabola `y^2 = 2x` such that a circle with PQ as diameter passes through the vertex O of the parabola. If P lies in the first quadrant and the area of the triangle `Delta OPQ` is `3 √2` , then which of the following is (are) the coordinates of `P?`

Text Solution

Verified by Experts

The correct Answer is:
A, D
Since, `angle POQ=90^(@)`

`rArr (t_(1)-0)/((t_(1)^(2))/(2)-0)*(t_(2)-0)/((t_(2)^(2))/(2)-0)= -1 rArr t_(1)t_(2)= -4 " ...(i)" `
`therefore ar(triangle OPQ)=3sqrt(2)`
` therefore (1)/(2)|(0,0,1),(t_(1)^(2)//2,t_(1),1),(t_(2)^(2)//2,t_(2),1)|=pm 3 sqrt(2) rArr (1)/(2)((t_(1)^(2)t_(2))/(2)-(t_(1)t_(2)^(2))/(2))=pm 3sqrt(2) `
`rArr (1)/(4)(-4t_(1)+4t_(2)) =pm 3sqrt(2) rArr t_(1)+(4)/(t_(1))=3 sqrt(2) [because t_(1)gt0 " for "P] `
`rArr t_(1)^(2)-3sqrt(2)t_(1)+4=0 rArr (t_(1)-2sqrt(2)(t_(1)-sqrt(2))=0 `
`rArr t_(1)=sqrt(2) " or " 2sqrt(2) `
` therefore P(1, sqrt(2)) " or " P(4,2sqrt(2))`
Promotional Banner

Similar Questions

Explore conceptually related problems

Let PQ be a chord of the parabola y^2=4x . A circle drawn with PQ as a diameter passes through the vertex V of theparabola. If ar(Delta PVQ)=20 sq unit then the coordinates of P are

Let A and B be two distinct points on the parabola y^2=4x . If the axis of the parabola touches a circle of radius r having AB as its diameter, then the slope of the line joining A and B can be

If chord BC subtends right angle at the vertex A of the parabola y^(2)=4x with AB=sqrt(5) then find the area of triangle ABC.

Length of the focal chord of the parabola (y +3)^(2) = -8(x-1) which lies at a distance 2 units from the vertex of the parabola is

The area (in sq. units) in the first quadrant bounded by the parabola y=x^2+1 , the tangent to it at the point (2, 5) and the coordinate axes is

T P and T Q are tangents to the parabola y^2=4a x at Pa n dQ , respectively. If the chord P Q passes through the fixed point (-a ,b), then find the locus of Tdot

The triangle PQR of area 'A' is inscribed in the parabola y^2=4ax such that the vertex P lies at the vertex pf the parabola and base QR is a focal chord.The modulus of the difference of the ordinates of the points Q and R is :

The point of intersection of the circle x^(2) + y^(2) = 8x and the parabola y^(2) = 4x which lies in the first quadrant is

An equilateral triangle is inscribed in the parabola y^(2) = 4ax whose vertex is at the vertex of the parabola. Find the length of its side.

IF the sides of the triangle are p , q sqrt(p^2 + q^2 + pq) then the largest angle is