Number of Common Tangents to Two Circles in Different Conditions

Q. The number of common tangents to the circles $x^2+y^2-4x-6y-12=0$ and $x^2+y^2+6x+18y+26=0$ is

1. 1
2. 2
3. 3
4. 4

Q. The centers of two circles $C_{1}and{C}_2$ each of unit radius are at a distance of 6 units from each other. Let P be the midpoint of the line segment joining the centers of $C_1$ and $C_2$ and $C$ be a circle touching circles $C_1$ and $C_2$ externally. If a common tangent to $C_{1}andC$ passing through P is also a common tangent to $C_{2}andC$, then find the radius of circle C.___

Q.

Let C be the circle with centre at (1, 1) and radius 1. If T is the circle centred at (0, y) passing through origin and touching the circle C externally, then the radius of T is equal to

1. $\frac{1}{2}$

2. $\frac{\sqrt{3}}{\sqrt{2}}$

3. $\frac{\sqrt{3}}{2}$

4. $\frac{1}{4}$

Q.

A ladder starts sliding on a wall from its initial position (fig 1) and (fig 2) is an intermediate stage
before it completely hits the ground. Find the locus of the curve traced by the mid-point of the ladder.
(a and b are constants).

1. 
2. 
3. 
4. 

Q. The centers of two circles $C_{1}and{C}_2$ each of unit radius are at a distance of 6 units from each other. Let P be the midpoint of the line segment joining the centers of $C_1$ and $C_2$ and $C$ be a circle touching circles $C_1$ and $C_2$ externally. If a common tangent to $C_{1}andC$ passing through P is also a common tangent to $C_{2}andC$, then find the radius of circle C.___