If the point $(1,4)$ lies inside the circle $x^2+y^2-6x-10y+p=0$ and the circle does not touch or intersect the coordinate axes,then

The line through $P(a, 2)$,where $a \neq 0$,making an angle $45^{\circ}$ with the positive direction of the $X$-axis meets the curve $\frac{x^2}{9}+\frac{y^2}{4}=1$ at $A$ and $D$ and the coordinate axes at $B$ and $C$. If $PA, PB, PC$ and $PD$ are in a geometric progression,then $2a=$

Two concentric circles are such that the smaller divides the larger into two regions of equal area. If the radius of the smaller circle is $2$,then the length of the tangent from any point $P$ on the larger circle to the smaller circle is:

The value of $c$ for which the set,$\{(x, y) | x^2 + y^2 + 2x \le 1 \} \cap \{(x, y) | x - y + c \ge 0\}$ contains only one point in common is :

Let $r_{1}$ and $r_{2}$ be the radii of the largest and smallest circles,respectively,which pass through the point $(-4, 1)$ and have their centres on the circumference of the circle $x^{2} + y^{2} + 2x + 4y - 4 = 0$. If $\frac{r_{1}}{r_{2}} = a + b \sqrt{2}$,then $a + b$ is equal to:

$A$ straight line passing through the point $(1,0)$ and not parallel to the $x$-axis intersects the curve $2x^2+5y^2-7x=0$ at two points $A$ and $B$. The angle subtended by the line segment $AB$ at the origin is (in $^\circ$)