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Question

For the circles S1x2+y24x6y12=0 and S2x2+y2+6x+4y12=0 and the line Lx+y=0

A
L is the common tangent of S1 and S2
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B
L is the common chord of S1 and S2
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C
L is radical axis of S1 and S2
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D
L is perpendicular to the line joining the centers of S1 and S2
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Solution

The correct options are
B L is radical axis of S1 and S2
C L is perpendicular to the line joining the centers of S1 and S2
D L is the common chord of S1 and S2
Centre of S1 is C1(2,3) and radius of S1 is r1=5.
Centre of S1 is C2(3,2)and radius of S1 is r1=5.
Also, d= distance between centres =C1C2=25+25=50
|r1r2|=0 and r1+r2=10.
Since |r1r2|<C1C2<r1+r2, therefore circles S1 and S2 intersect.
Equation of the common chord of C1 and C21 is S1S2=0 i.e., x+y=0,
which is also the equation of the radical axis and hence it is to the line joining the centres C1 and C2 of the two circles.

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