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 sqrt 2` , then which of the following is (are) the coordinates of `P?` (a) ( 4 , 2 √ 2 ) (b) ( 9 , 3 √ 2 ) (c) ( 1 4 , 1 √ 2 ) (d) ( 1 , √ 2 )

A

`(4,2sqrt(2))`

B

`(9,3sqrt(2))`

C

`((1)/(4),(1)/(sqrt(2)))`

D

`(1,sqrt(2))`

Text Solution

Verified by Experts

The correct Answer is:
A, D
1,4
`OPbotOQ`
`rArrt_(1)t_(2)=-4`
`rArr" Area of "DeltaOPQ=(1)/(2)OP*OQ`

`rArr|(1)/(2)sqrt((t_(1)^(4))/(4)+t_(1)^(2))sqrt((t_(2)^(4))/(4)+t_(2)^(2))|=3sqrt(2)`
`rArr*4*sqrt(((t_(1)^4+4))/(4)((t_(2)^(2)+4))/(4))=3sqrt(2)`
`rArr4*((16+4(t_(1)^(2)+t_(2)^(2))+16))/(16)=9xx2`
`rArr8+t_(1)^(2)+t_(2)^(2)=18`
`rArrt_(1)^(2)+t_(2)^(2)-10=0`
`rArrt_(1)^(4)-10t_(1)^(2)+16=0`
`rArrt_(1)^(2)=2,8`
Promotional Banner

Topper's Solved these Questions

  • PARABOLA

    CENGAGE PUBLICATION|Exercise Comprehension Type|2 Videos

  • PARABOLA

    CENGAGE PUBLICATION|Exercise Single Correct Answer Type|46 Videos

  • PARABOLA

    CENGAGE PUBLICATION|Exercise JEE ADVENCED SINGLE CORRECT ANSWER TYPE|2 Videos

  • PAIR OF STRAIGHT LINES

    CENGAGE PUBLICATION|Exercise Numberical Value Type|5 Videos

  • PERMUTATION AND COMBINATION

    CENGAGE PUBLICATION|Exercise Comprehension|8 Videos

Similar Questions

Explore conceptually related problems

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 OPQ is 3 sqrt(2) , then which of the following is (are) the coordinates of P ?

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 the parabola. If ar(Delta PVQ)=20 sq unit then the coordinates of P are

Find the point on the parabola y^(2)=4ax(a gt 0) which forms a triangle of area 3a^(2) with the vertex and focus of the parabola .

Let A(x_(1),y_(1)) and B(x_(2),y_(2)) be two points on the parabola y^(2) = 4ax . If the circle with chord AB as a dimater touches the parabola, then |y_(1)-y_(2)| is equal to

The coordinates of the vertex of the parabola (x + 1)^(2) =- 9(y +2) are

The triangle P Q R of area A is inscribed in the parabola y^2=4a x such that the vertex P lies at the vertex of the parabola and the base Q R is a focal chord. The modulus of the difference of the ordinates of the points Qa n dR is

An equilateral triangle inscribe a parabola y^2 = 4ax . such that oneof the vertex of the triangle lies on the vertex of the parabola-. Find the,length of,the each side. of the triangle:

If (a ,b) is the midpoint of a chord passing through the vertex of the parabola y^2=4(x+1), then prove that 2(a+1)=b^2

Through the vertex O of the parabola y^(2) = 4ax , a perpendicular is drawn to any tangent meeting it at P and the parabola at Q. Then OP, 2a and OQ are in

P and Q are any two points on the circle x^2+y^2= 4 such that PQ is a diameter. If alpha and beta are the lengths of perpendiculars from P and Q on x + y = 1 then the maximum value of alphabeta is