Number of Common Tangents to Two Circles in Different Conditions

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

Q.

Let the tangents drawn from the origin to the circle, $x^2+y^2-8x-4y+16=0$ touch it at the points $A$ and $B$. The ${\left(AB\right)}^2$ is equal to

1. $\frac{32}{5}$

2. $\frac{64}{5}$

3. $\frac{52}{5}$

4. $\frac{56}{5}$

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. The common tangent to the circles $x^2+y^2=4$ and $x^2+y^2+6x+8y-24=0$ also passes through the point :

1. $(-6,4)$
2. $(6,-2)$
3. $(4,-2)$
4. $(-4,6)$

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 CORRECT combination?

1. $\left(I\right),\left(S\right)$
2. $\left(I\right),\left(U\right)$
3. $\left(II\right),\left(Q\right)$
4. $\left(II\right),\left(T\right)$

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. $\frac{1}{4}$

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

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

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

Q. If two circles $x^2+y^2-2ax+c^2=0$ and $x^2+y^2-2by+c^2=0$ touch each other externally, then

1. $\frac{1}{a^2}+\frac{1}{c^2}=\frac{1}{b^2}$
2. $\frac{1}{a^2}+\frac{1}{b^2}=\frac{1}{c^2}$
3. $\frac{1}{a^2}+\frac{1}{b^2}=\frac{2}{c^2}$
4. $\frac{1}{b^2}+\frac{1}{c^2}=\frac{1}{a^2}$

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. The number of common tangents to the circles $x^2+y^2-4x-2y+1=0$ and $x^2+y^2-6x-4y+4=0$ is

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. If the circles $x^2+y^2=9$ and $x^2+y^2-8x-6y+n^2=0,n\in Z$ have exactly two common tangents, then the number of possible values of $n$ is

Q. The number of common tangents to the following pairs of circles $x^2+y^2=4,x^2+y^2-6x-8y+16=0$ is

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. 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. The minimum value of $|\left[\mu \right]|$ for which the circle $x^2+y^2=9$ and $x^2+y^2-\mu x-6=0$ have two common tangents is
(where [.] is greatest integer function)

Q. The number of common tangents to the circles $x^2+y^2+6x+6y+14=0$ and $x^2+y^2-2x-4y-4=0$ is

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

Q. The equation of the circle which passes through $(8,16)$ and touches the line $4x-3y=64$ at $(16,0)$ is

1. $x^2+y^2-16x-12y=0$
2. $x^2+y^2-12x-16y=100$
3. $x^2+y^2-12x-16y=0$
4. $x^2+y^2-16x-12y=100$

Q. The common chord of the circle $x^2+y^2+6x+8y-7=0$ and another circle passing through origin, which touches the line y = x, always passes through the point $(\alpha ,\beta )$, then $|\alpha -\beta |$=___

Q. The number of common tangents to the circles $x^2+y^2-4x-6y-3=0$ and $x^2+y^2+2x+2y+1=0$ is

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

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 CORRECT combination?

1. $\left(I\right),\left(S\right)$
2. $\left(I\right),\left(U\right)$
3. $\left(II\right),\left(Q\right)$
4. $\left(II\right),\left(T\right)$

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. 2
2. 4
3. 3
4. 1

Q. Which of the following point lies on the common tangent at the circle $x^2+y^2-4x-6y-36=0$ and $x^2+y^2-10x+2y+22=0$.

1. $(10,\frac{1}{4})$
2. $(\frac{1}{4},10)$
3. $(12,2)$
4. $(2,12)$

Q. The number of common tangents to the following pairs of circles $x^2+y^2+4x-6y-3=0$ and $x^2+y^2+4x-2y+4=0$ is

1. $0$
2. $1$
3. $2$
4. $3$

Q. The number of common tangents to the circles $x^2+y^2+6x+6y+14=0$ and $x^2+y^2-2x-4y-4=0$ is

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

Q. The locus of the centre of a circle touching the lines $2x+3y–2=0\text{and}2x–3y+2=0$ may be

1. $x=0$
2. $y=\frac{2}{3}$
3. $3xy+2x=0$
4. $3xy-2x=0$

Q. The equations of circles with radius $3$ units and touching the circle $x^2+y^2-2x-4y-20=0$ at $(5,5)$ is/are

1. $(5x-16{)}^2+(5y-13{)}^2=225$
2. $(5x-13{)}^2+(5y-16{)}^2=225$
3. $(5x-34{)}^2+(5y-37{)}^2=225$
4. $(5x-37{)}^2+(5y-34{)}^2=225$

Q. Find the centre and radius of the circle $x^2+y^2-8x+10y-12=0$.

1. $(4,-5),\sqrt{53}$
2. $(-4,-5),\sqrt{53}$
3. $(-4,5),\sqrt{53}$
4. $(4,5),\sqrt{53}$

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

1. $3$
2. $2$
3. $5$
4. $4$

Q. A circle passes through the points A(1, 0), B(5, 0) and C(0, h). If $\angle ACB$ is maximum then

1. 
2. 
3. 
4. 

Q. If the circle $x^2+y^2+2ax+c=0$ and $x^2+y^2+2by+c=0$ touch each other, then

1. $\frac{1}{a^2}+\frac{1}{b^2}=\frac{1}{c}$
2. $\frac{1}{a^2}+\frac{1}{b^2}=\frac{1}{c^2}$
3. $a+b=2c$
4. $\frac{1}{a}+\frac{1}{b}=\frac{2}{c}$