If two circles of the same radius $a$ and centers at $(2, 3)$ and $(5, 6)$ are orthogonal,find the value of $a$.

$B$ 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:

$A$ common tangent to $9x^2 + 16y^2 = 144$,$y^2 - x + 4 = 0$,and $x^2 + y^2 - 12x + 32 = 0$ 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 $L_1$?

For the circle $C$ with the equation $x^2+y^2-16x-12y+64=0$,match the List-$I$ with the List-$II$ given below.

List-$I$	List-$II$
$(i)$ The equation of the polar of $(-5, 1)$ with respect to $C$	$(A)$ $y = 0$
$(ii)$ The equation of the tangent at $(8, 0)$ to $C$	$(B)$ $y = 6$
$(iii)$ The equation of the normal at $(2, 6)$ to $C$	$(C)$ $x + y = 7$
$(iv)$ The equation of the diameter of $C$ through $(8, 12)$	$(D)$ $13x + 5y = 98$
	$(E)$ $x = 8$

The correct match is:

If the curves $ax^2+by^2=1$ and $cx^2+dy^2=1$ intersect orthogonally, then $\frac{b-a}{d-c}=$