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 number of common tangents to the circles $x^2+y^2+2x+8y-23=0\mathrm{and}{x}^2+y^2-4x-10y+9=0$ are .

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

Q. For circles $x^2+y^2+2x-8y+13=0$ and $x^2+y^2-12x-14y+76=0$ equation of all the common tangents are:

1. $y+2=\frac{21\pm \sqrt{57}}{48}(x+15)$
2. $y-\frac{26}{5}=\frac{21\pm 5\sqrt{33}}{24}(x-\frac{9}{5})$
3. $y-\frac{26}{5}=\frac{21\pm 5\sqrt{33}}{24}(x-\frac{10}{5})$
4. $y-2=\frac{21\pm \sqrt{57}}{48}(x-15)$

Q.

The two circles $x^2+y^2-4y=0$ and $x^2+y^2-8y=0$

1. Touch each other internally

2. Touch each other externally

3. Do not touch each other

4. intersect each other

Q. If the circles $x^2+y^2-2x-4y=0$ and $x^2+y^2-8y-k=0$ touch each other internally, then the value of $k$ is

1. $-16$
2. $4$
3. $3$
4. $-2$

Q.

Circles $x^2+y^2-2x-4y=0and{x}^2+y^2-8y-4=0$

1. Touch each other internally

2. Cuts each other at two points

3. Touch each other externally

4. One circle lies inside the other

Q. Let the circles $C_1:x^2+y^2=9$ and $C_2:(x-3{)}^2+(y-4{)}^2=16,$ intersect at the points $X$ and $Y.$ Suppose that another circle $C_3:(x-h{)}^2+(y-k{)}^2=r^2$ satisfies the following conditions:

$\left(i\right)$ centre of $C_3$ is collinear with the centres of $C_1$ and $C_2.$

$\left(ii\right)$$C_1$ and $C_2$ both lie inside $C_3$, and

$\left(iii\right)$ $C_3$ touches $C_1$ at $M$ and $C_2$ at $N$

Let the line through $X$ and $Y$ intersect $C_3$ at $Z$ and $W$, and let a common tangent of $C_1$ and $C_3$ be the tangent to the parabola $x^2=8\alpha y.$
There are some expressions given in $List-I$ whose values are given in $List-II$ below:

$\begin{array}{cccc}& \text{List}I& & \text{List}II\\ \left(I\right)& 2h+k& \left(P\right)& 6\\ \left(II\right)& \frac{\text{length of}ZW}{\text{length of}XY}& \left(Q\right)& \sqrt{6}\\ \left(III\right)& \frac{\text{Area of triangle}MZN}{\text{Area of triangle}ZMW}& \left(R\right)& \frac{5}{4}\\ \left(IV\right)& \alpha & \left(S\right)& \frac{21}{5}\\ & & \left(T\right)& 2\sqrt{6}\\ & & \left(U\right)& \frac{10}{3}\end{array}$

Which of the following is the only INCORRECT combination?

1. $\left(I\right),\left(P\right)$
2. $\left(IV\right),\left(U\right)$
3. $\left(III\right),\left(R\right)$
4. $\left(IV\right),\left(S\right)$

Q. If two circles of radii $5$ units touches each other at $(1,2)$ and the equation of the common tangent is $4x+3y=10$, then the equation of the circle is/are

1. $x^2+y^2-10x-10y+25=0$
2. $x^2+y^2+6x+2y-15=0$
3. $x^2+y^2-10x-10y-25=0$
4. $x^2+y^2+6x-2y+15=0$

Q. If the circles $x^2+y^2-16x-20y+164=r^2$ and $(x-4{)}^2+(y-7{)}^2=36$ intersect at two distinct points, then :

1. $1<r<11$
2. $0<r<1$
3. $r=11$
4. $r>11$

Q.

The number of common tangents to the circles $x^2+y^2+2x+8y-23=0and{x}^2+y^2-4x-10y+19=0$ is

1. 1

2. 2

3. 3

4. 4

Q. If $3x+5y+17=0$ is polar for the circle $x^2+y^2+4x+6y+9=0$, then the pole is

1. $(1,2)$
2. $(-1,2)$
3. $(2,1)$
4. $(1,-2)$

Q. A circle is given by $x^2+(y-1{)}^2=1$, another circle $C$ touches it externally and also the $x$-axis, then the locus of its centre is

1. $x^2=2y$
2. $x^2=3y$
3. $x^2=y$
4. $x^2=4y$

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. $\frac{1}{4}$

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

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