Let $AB$ be a chord of the circle $x^2 + y^2 = r^2$ subtending a right angle at the centre. Then the locus of the centroid of the $\Delta PAB$ as $P$ moves on the circle is

If the line $ax + by = 0$ touches the circle ${x^2} + {y^2} + 2x + 4y = 0$ and is a normal to the circle ${x^2} + {y^2} - 4x + 2y - 3 = 0$,then the value of $(a, b)$ will be

The shortest distance between the curves $y^2=8x$ and $x^2+y^2+12y+35=0$ is:

$A$ circle passes through the point $\left( 3, \sqrt{\frac{7}{2}} \right)$ and touches the line pair $x^2 - y^2 - 2x + 1 = 0$. The coordinates of the centre of the circle are:

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:

$A(x_1, y_1)$ is the internal centre of similitude and $B(x_2, y_2)$ is the external centre of similitude of two circles $C_1$ and $C_2$ whose centres are $P(\alpha, \beta)$ and $Q(\gamma, \delta)$ respectively. If $PA=3, AB=5, QB=2$,then the ratio of the radii of the two circles is: