CircleEasy
Question
Tangents OP and OQ are drawn from the origin O to the circle x2 + y2 + 2gx + 2fy + c = 0. Then, the equation of the circumcircle of the triangle OPQ is :-
Options
A.x2 + y2 + 2gx + 2fy = 0
B.x2 + y2 + gx + fy = 0
C.x2 + y2 - gx - fy = 0
D.x2 + y2 - 2gx - 2fy = 0
Solution
The equation of the chord of contact of tangents drawn from the origin to the circle
x2 + y2 + 2gx + 2fy = c = 0 is gx + fy + c = 0 ...(i)
The required circle passes through the intersection of the given circle and line (i).
Therefore, its equation is
(x2 + y2 + 2gx + 2fy + c) + λ (gx + fy + c) = 0 ...(ii)
this passes through (0, 0)
∴ c + λc = 0 ⇒ λ = - 1
putting λ = -1 in (ii), the eq. of the req. circle is x2 + y2 + gx + fy = 0.
x2 + y2 + 2gx + 2fy = c = 0 is gx + fy + c = 0 ...(i)
The required circle passes through the intersection of the given circle and line (i).
Therefore, its equation is
(x2 + y2 + 2gx + 2fy + c) + λ (gx + fy + c) = 0 ...(ii)
this passes through (0, 0)
∴ c + λc = 0 ⇒ λ = - 1
putting λ = -1 in (ii), the eq. of the req. circle is x2 + y2 + gx + fy = 0.
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