Direct Common Tangent

Q. For any two circles, the number of direct common tangents will always be greater than number of transverse common tangents.

1. True
2. False

Q. The circles $x^2+y^2-2x-4y+1=0$ and $x^2+y^2+4x+4y-1=0$

1. touch internally
2. touch externally
3. have $3x+4y-1=0$ as the common tangent at the point of contact.
4. have $3x+4y+1=0$ as the common tangent at the point of contact.

Q.

Which of the following statements is/are correct?

1. With respect to a tangent both the circles lie on the same side, this tangent is called direct common tangent.

2. With respect to a tangent both the circles lie on the opposite side, this tangent is called transverse (indirect) common tangent.

1. Only 1

2. Only 2

3. Both 1 & 2

4. None of these

Q. A variable circle always touches the line $y=x$ at the origin. If all the common chords of the given circle and $x^2+y^2+6x+8y-7=0$ passes through a fixed point $(a,b)$, then $a+b$ is

Q. Three circles of radii $a,b,c(a<b<c)$ touch each other externally. If they have $x$-axis as a common tangent, then:

1. $a,b,c$ are in A.P.
2. $\sqrt{a},\sqrt{b},\sqrt{c}$ are in A.P.
3. $\frac{1}{\sqrt{a}}=\frac{1}{\sqrt{b}}+\frac{1}{\sqrt{c}}$
4. $\frac{1}{\sqrt{b}}=\frac{1}{\sqrt{a}}+\frac{1}{\sqrt{c}}$

Q. If circles with radii $a$ units and $b$ units touch each other externally and the angle between their direct common tangents is $\theta$, where $a>b\ge 2$, then the value of $\sin \theta$ is

1. $\frac{2(a-b)\left(\sqrt{ab}\right)}{(a+b{)}^2}$
2. $\frac{2(a+b)\left(\sqrt{ab}\right)}{(a-b{)}^2}$
3. $\frac{4(a-b)\left(\sqrt{ab}\right)}{(a+b{)}^2}$
4. $\frac{4(a+b)\left(\sqrt{ab}\right)}{(a-b{)}^2}$

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.

The equation of a circle of radius 1 touching the circles $x^2+y^2-2|x|=0$ is

1. $x^2+y^2+2\sqrt{3}x-2=0$

2. $x^2+y^2-2\sqrt{3}y-2=0$

3. $x^2+y^2+2\sqrt{3}y+2=0$

4. $x^2+y^2+2\sqrt{3}x+2=0$