Home
Class 12
MATHS
The circle C1 : x^2 + y^2 = 3, with cent...

The circle C1 :` x^2 + y^2 = 3`, with center at O, intersects the parabola` x^2 = 2y` at the point P in the first quadrant. Let the tangent to the circle C1 at P touches other two circles C2 and C3 at R2 and R3, respectively. Suppose C2 and C3 have equal radii `2sqrt(3)` and centers Q2 and Q3, respectively.If `Q_2 and Q_3` lies on the y-axis, then find `Q_(2)Q_(3)=12` and `R_(2)R_(3)=4sqrt(6)`

Text Solution

Verified by Experts

The correct Answer is:
A, B, C
1,2,3
`x^(2)=y^(2)=3andx^(2)=2y`
Solving above curves, we get point of intersection as `P-=(sqrt(2),1)`.
Equation of tangent to circle at this point `sqrt(2x)+y=3`.
`:." "tantheta=-sqrt(2)`
`tanalpha=tan(theta-90^(@))=-cottheta=(1)/(sqrt(2))`

So, `sinalpha=(1)/(sqrt(3))=(Q_(3)R_(3))/(Q_(3)T)=(2sqrt(3))/(Q_(3)T)`
`rArrQ_(3)T=6`
`:." "Q_(2)Q_(3)=2Q_(3)T=12` (As circle have equal radii)
`tanalpha=(1)/(sqrt(2))=(Q_(3)R_(3))/(R_(3)T)=(2sqrt(3))/(R_(3)T)`
`rArrR_(3)T=2sqrt(6)`
`rArrR_(2)R_(3)=2R_(3)T=4sqrt(6)`
Perpendiculau distance of O from `R_(2)R_(3)=|(3)/(sqrt((sqrt(2))^(2)+1^(2)))|=sqrt(3)`
`:." Area"(OR_(2)R_(3))=(1)/(2)xxsqrt(3)xx4sqrt(6)=6sqrt(2)` sq. units
Similarly, Area `(PQ_(2)Q_(3))=(1)/(2)xxsqrt(2)xx12=6sqrt(2)` sq. units
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

If the line y=3x+c touches the parabola y^2=12 x at point P , then find the equation of the tangent at point Q where P Q is a focal chord.

If the straight line x - 2y + 1 = 0 intersects the circle x^2 + y^2 = 25 at points P and Q, then find the coordinates of the point of intersection of the tangents drawn at P and Q to the circle x^2 + y^2 = 25 .

If line x-2y-1=0 intersects parabola y^(2)=4x at P and Q, then find the point of intersection of normals at P and Q.

If the circle x^2+y^2=a^2 intersects the hyperbola x y=c^2 at four points P(x_1, y_1),Q(x_2, y_2),R(x_3, y_3), and S(x_4, y_4), then

Let circle C_1 : x^2 + (y-4)^2 = 12 intersects circle C_2: (x-3)^2 +y^2=13 at A and B. A quadrilateral ACBD is formed by tangents at A and B to both circles. The diameter of circumcircle of quadrilateral ACBD is

If tangent at (1, 2) to the circle C_1: x^2+y^2= 5 intersects the circle C_2: x^2 + y^2 = 9 at A and B and tangents at A and B to the second circle meet at point C, then the co- ordinates of C are given by

The circle x^2+y^2=5 meets the parabola y^2=4x at P and Q . Then the length P Q is equal to (a)2 (b) 2sqrt(2) (c) 4 (d) none of these

The common tangents to the circle x^2 + y^2 =2 and the parabola y^2 = 8x touch the circle at P,Q andthe parabola at R,S . Then area of quadrilateral PQRS is

Find the points of contact Q and R of a tangent from the point P(2,3) on the parabola y^2=4xdot