Two tangents are drawn from a point $P$ on radical axis to the two circles touching at $Q$ and $R$ respectively then triangle formed by joining $PQR$ is

The locus of centre of the circle which cuts the circles${x^2} + {y^2} + 2{g_1}x + 2{f_1}y + {c_1} = 0$ and ${x^2} + {y^2} + 2{g_2}x + 2{f_2}y + {c_2} = 0$ orthogonally is

Coordinates of the centre of the circle which bisects the circumferences of the circles

$x^2 + y^2 = 1 ; x^2 + y^2 + 2x - 3 = 0$ and $x^2 + y^2 + 2y - 3 = 0$ is

The condition of the curves $a{x^2} + b{y^2} = 1$and $a'{x^2} + b'{y^2} = 1$ to intersect each other orthogonally, is

Let $C_1$ be the circle of radius $1$ with center at the origin. Let $C_2$ be the circle of radius $\mathrm{I}$ with center at the point $A=(4,1)$, where $1<\mathrm{r}<3$. Two distinct common tangents $P Q$ and $S T$ of $C_1$ and $C_2$ are drawn. The tangent $P Q$ touches $C_1$ at $P$ and $C_2$ at $Q$. The tangent $S T$ touches $C_1$ at $S$ and $C_2$ at $T$. Mid points of the line segments $P Q$ and $S T$ are joined to form a line which meets the $x$-axis at a point $B$. If $A B=\sqrt{5}$, then the value of $r^2$ is

The condition that the circle ${(x - 3)^2} + {(y - 4)^2} = {r^2}$ lies entirely within the circle ${x^2} + {y^2} = {R^2},$ is