Number of Common Tangents to Two Circles in Different Conditions

Q.

The equation of the tangent to the circle $x^2+y^2+4x-4y+4=0$ which makes equal intercepts on the positive coordinates axes, is

1. $x+y=2$

2. $x+y=2\sqrt{2}$

3. $x+y=4$

4. $x+y=8$

Q. The circles $x^2+y^2-2x-4y=0$ and $x^2+y^2-8y-4=0$ have

1. $2$ common tangents.
2. $3$ common tangents.
3. $4$ common tangents.
4. $1$ common tangent.

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

Q. If the lengths of the tangents from two points A, B to a circle are 6, 7 respectively. If A, B are conjugate points then AB =

1. 5
2. $\sqrt{85}$
3. $\frac{\sqrt{85}}{2}$
4. none

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. 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. $-2$
3. $3$
4. $4$

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.

The two circles $x^2+y^2+2ax+c=0and{x}^2+y^2+2by+c=0$ touch if $\frac{1}{a^2}+\frac{1}{b^2}=$

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

2. $c$

3. $\frac{1}{c}$

4. $c^2$

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

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

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