    Question

# The circle C1:x2+y2=3, with centre at O, intersect the parabola x2=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 (2√3 ) and centres Q2 and Q3 respectively. If Q2 and Q3 lie on the y-axis, then:

A
Q2Q3=12
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B
R2R3=46
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C
Area of the triangle OR2R3 is 62
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D
Area of the triangle PQ2Q3 is 42
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Solution

## The correct options are A Q2Q3=12 B R2R3=4√6 C Area of the triangle OR2R3 is 6√2The point of intersection of the curves x2+y2=3 and x2=2y is (√2,1) The tangent at the point (√2,1) is given by: √2x+y=3 Let Q2 or Q3 be (0,k) Then, |√2(0)+k−3|√3=2√3 k=9 or−3 Q2(0,9) and Q3(0,−3) Q2Q3=12 R2R3=√(Q2Q3)2−(r2+r3)2 (where r2 & r3 are the radius of Q2 & Q3 respectively) R2R3=√(12)2−(4√3)2=4√6 For triangle OR2R3, Let h be the length of perpendicular from O to the side R2R3 h=√3 Area of OR2R3=12h(R2R3)=6√2 Area of PQ2Q3=12×12×√2=6√2  Suggest Corrections  0      Related Videos   Tangent to a Circle
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