Equation of Family of Circles Passing through Points of Intersection of Circle and a Line

Q. If a circle $C$ passing through the point $(4,0)$ touches the circle $x^2+y^2+4x-6y=12$ externally at the point $(1,-1)$ then the radius of $C$ is :

1. $4$
2. $2\sqrt{5}$
3. $\sqrt{57}$
4. $5$

Q. The line $x=y$ touches a circle at the point $(1,1).$ If the circle also passes through the point $(1,-3),$ then its radius(in units) is :

1. $2$
2. $3$
3. $2\sqrt{2}$
4. $3\sqrt{2}$

Q. The equation of the circle passing through $(1,1)$ and the points of intersection of $x^2+y^2+13x-3y=0$ and $11x+\frac{1}{2}y+\frac{25}{2}=0\text{is}S$, then point $(2,2)$ is

1. lying on the circle $S$
2. centre of circle $S$
3. lying inside the circle $S$
4. lying outside the circle $S$

Q. If the line $2x-y+1=0$ is a tangent to the circle $S\equiv 0$ at $P(2,5)$ and the centre of the circle lies on $x-2y=4,$ then the intercept made by the circle $S\equiv 0$ on the $x-$axis is equal to

1. $2\sqrt{23}$
2. $2\sqrt{41}$
3. $2\sqrt{35}$
4. $2\sqrt{63}$

Q. Tangent to the ellipse $\frac{x^2}{4}+y^2=1$ at the point $P(\sqrt{2},\frac{1}{\sqrt{2}})$ touches the circle $x^2+y^2=r^2$ at the point $Q.$ Then the length of $PQ$ is

1. $\frac{1}{\sqrt{10}}$
2. $\frac{11}{\sqrt{10}}$
3. $\frac{3}{\sqrt{10}}$
4. $\frac{7}{\sqrt{10}}$

Q. For the four circles $M,N,O$ and $P,$ following four equations are given:
Circle $M:x^2+y^2=1$
Circle $N:x^2+y^2–2x=0$
Circle $O:x^2+y^2–2x–2y+1=0$
Circle $P:x^2+y^2–2y=0$
If the centre of circle $M$ is joined with centre of the circle $N,$ further centre of circle $N$ is joined with centre of the circle $O,$ centre of circle $O$ is joined with the centre of circle $P$ and lastly, centre of circle $P$ is joined with centre of circle $M,$ then these lines form the sides of a :

1. Rectangle
2. Square
3. Parallelogram
4. Rhombus

Q. Find the equation of the circle passing through the points
$(1,2)(3,-4)$ and $(5,-6)$

Q. Find $|z|$, if $z=5-12i:$

1. $17$
2. $13$
3. Both (a) and (b)
4. $-13$

Q. The equation of the circle passing through $(1,1)$ and the points of intersection of $x^2+y^2+13x-3y=0$ and $11x+\frac{1}{2}y+\frac{25}{2}=0$ is $S.$ Then point $(2,2)$ is

1. lying on the circle $S$
2. centre of circle $S$
3. lying inside the circle $S$
4. lying outside the circle $S$

Q.

$\text{If}|z-\frac{4}{z}|=2,\text{then the greatest value of}|z|\text{is}$

1. $\sqrt{5}+1$

2. $\sqrt{3}+1$

3. $2+\sqrt{2}$

4. $1+\sqrt{2}$

Q. Circle(s) touching x-axis at a distance 3 from the origin and having an intercept of length $2\sqrt{7}$ on y-axis is (are)

1. $x^2+y^2-6x+7y+9=0$
2. $x^2+y^2-6x-8y+9=0$
3. $x^2+y^2-6x+8y+9=0$
4. $x^2+y^2-6x-7y+9=0$

Q.

Consider the family of circles $x^2+y^2-2x-2\lambda y-8=0$ passing through two fixed points A and B. Then the distance between the points A and B, is $\underset{–}{}$

1. 6

2. $\sqrt{6}$

3. $\sqrt{18}$

4. 36

Q. If a circle $C$ passing through the point $(4,0)$ touches the circle $x^2+y^2+4x-6y=12$ externally at the point $(1,-1)$ then the radius of $C$ is :

1. $\sqrt{57}$
2. $4$
3. $2\sqrt{5}$
4. $5$

Q. Let $x,y,z\in C$ satisfy $|x|=1,|y-6-8i|=3$ and $|z+1-7i|=5$ respectively, then the minimum value of $|x-z|+|y-z|$ is equal to

1. $2$
2. $5$
3. $1$
4. $6$

Q. If a circle $S(x,y)=0$ touches at the point $(2,3)$ of the line $x+y=5$ and $S(1,2)=–2,$ then radius of such circle

1. $2$ units
2. $4$ units
3. $\frac{1}{2}$ units
4. $\frac{1}{\sqrt{2}}$units

Q.

Consider the family of circles $x^2+y^2-2x-2\lambda y-8=0$ passing through two fixed points A and B. Then the distance between the points A and B, is $\underset{–}{}$

1. 36

2. 

3. 6

4.