If the distances from the origin to the centres of the three circles $x^2 + y^2 - 2\lambda_i x = c^2$ for $i = 1, 2, 3$ are in $G.P.$,then the lengths of the tangents drawn to them from any point on the circle $x^2 + y^2 = c^2$ are in

Each of the two orthogonal circles $C_1$ and $C_2$ passes through both the points $(2,0)$ and $(-2,0)$. If $y=mx+c$ is a common tangent to these circles,then

Let $P(3 \cos \alpha, 2 \sin \alpha)$,$\alpha \neq 0$,be a point on the ellipse $\frac{x^2}{9} + \frac{y^2}{4} = 1$,$Q$ be a point on the circle $x^2 + y^2 - 14x - 14y + 82 = 0$,and $R$ be a point on the line $x + y = 5$ such that the centroid of the triangle $PQR$ is $(2 + \cos \alpha, 3 + \frac{2}{3} \sin \alpha)$. Then the sum of the ordinates of all possible points $R$ is:

$B$ and $C$ are fixed points having coordinates $(3, 0)$ and $(-3, 0)$ respectively. If the vertical angle $\angle BAC$ is $90^o$,then the locus of the centroid of the $\Delta ABC$ has the equation:

The locus of the midpoints of the chords of the circle $x^2 + y^2 + 4x - 6y - 12 = 0$ which subtend an angle of $\frac{\pi}{3}$ radians at its circumference is:

Let $L_1$ be a straight line passing through the origin and $L_2$ be the straight line $x + y = 1$. If the intercepts made by the circle $x^2 + y^2 - x + 3y = 0$ on $L_1$ and $L_2$ are equal,then which of the following equations can represent $L_1$?