Radical axis

LOL' and MOM' are two chords of parabola y^(2)=4ax with vertex A passing through a point O on its axis.Prove that the radical axis of the circles described on LL' and MM' as diameters passes though the vertex of the parabola.

The circle x^(2)+y^(2)-8x-10y+37=0 after being reflected about the line y=x, is shifted 2sqrt(2) units towards the origin,parallel to this line.Then the equation of the radical axis of the given circle and the new circle is

Through a point P(-2,0), tangerts PQ and PR are drawn to the parabola y^(2)=8x. Two circles each passing though the focus of the parabola and one touching at Q and other at R are drawn.Which of the following point(s) with respect to the triangle PQR lie(s) on the radical axis of the two circles?

Let two tangents be drawn to the ellipse (x^(2))/(4)+(y^(2))/(3)=1 from the point P(4,sqrt(3)) which touches the ellipse at Q and R. If equation of the radical axis of two cireles drawn PQ and PR as diameters is ax-by=2 then find the value of (b^(2)-a^(2))

Consider points A(sqrt13,0) and B(2sqrt13,0) lying on x-axis. These points are rotated anticlockwise direction about the origin through an angle of tan^-1(2/3). Let the new position of A and B be A' and B' respectively. With A' as centre and radius 2sqrt13/3 a circle C_1 is drawn and with B' as centre and radius sqrt13/3 circle C_2, is drawn. The radical axis of C_1 and C_2 is

For the circles S_1: x^2 + y^2-4x-6y-12 = 0 and S_2 : x^2 + y^2 + 6x + 4y-12=0 and the line L.:x+y=0 (A) L is common tangent of S_1 and S_2 (B) L is common chord of S_1 and S_2 (C) L is radical axis of S_1 and S_2 (D) L is perpendicular to the line joining the cente of S_1 & S_2