CircleHard
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.
Create a free account to view solution
View Solution FreeMore Circle Questions
The point diametrically opposite to the point P(1, 0) on the circle x2 + y2 + 2x + 4y - 3 = 0 is...The eccentric angle of the point where the line, 5x - 3y = 8 √2 is a normal to the ellipse = 1 is -...The common chord of two intersecting circles C1 and C2 can be seen from their centres at the angles of 90o & 60o respect...In the argand plane the inequality |(√3 + i)z - (√2 - i)|2 + |(√ + i)z + (√3 - i) |2...The equation of the common tangent touching the circle (x -3)2 + y2 = 9 and the parabola y2 = 4x above the x-axis is...