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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 circle C2 and C3 at R2 and R3, respectively. Suppose C2 and C3 have equal radii 23 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 trianlge PQ2Q3 is 42
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Solution

The correct options are
A Q2Q3=12
B Area of the triangle OR2R3 is 62
D R2R3=46
x2+y2=3,x2=2yy2+2y3=0y=1,x=2
Tangent at (2,1) to C1 is 2x+y=3
Let centre of circle touching this tangent is (0,y)
Hence, 2(0)+y32+1=23
|y3|=6y3=±6y=3,9
Hence, Q2(0,9),Q3(0,3),Q2Q3=12
Now C2:x2+(y9)2=(23)2;C3:x2+(y+3)2=(23)2
Now foot of perpendicular from Q2 and Q3 on tangent
x02=y91=(0+933)=2x=22,y=7,R2(22,7)
Similarly, x02=y+31=(0333)=2x=22,y=1,R3(22,1)
Hence, R2R3==(42)2+(8)2=46
Area of ΔOR2R3=12∣ ∣ ∣00122712211∣ ∣ ∣=12[1(227.22)]=62

Area of ΔPQ2Q3=12∣ ∣ ∣211091031∣ ∣ ∣=122(9+3)=62

516333_478062_ans.JPG

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