Equation of Family of Circles Touching a Line and Passing through a Given Point on the Line

Q. Five circles $C_1,C_2,C_3,C_4,C_5$ with radii $r_1,r_2,r_3,r_4,r_5$ respectively $(r_1<r_2<r_3<r_4<r_5)$ be such that $C_i$ and $C_i+1$ touch each other externally for all $i=1,2,3,4$. If all the five circles touches two straight lines $L_1$ and $L_2$ and $r_1=2$ and $r_5=32$, then $r_3$ is(units)

Q.

The other end of the diameter through the point $\left(-1,1\right)$ on the circle $x^2+y^2-6x+4y-12=0$ is

1. $\left(-7,5\right)$

2. $\left(-7,-5\right)$

3. $\left(7,-5\right)$

4. $\left(7,5\right)$

Q. If the lines $x-2y+3=0$ and $3x+ky+7=0$ cut the coordinate axes in concyclic points, then

1. $k=\frac{3}{2}$
2. $k=-\frac{3}{2}$
3. Equation of circle passing through those points is ${(x+\frac{8}{3})}^2+{(y-\frac{37}{12})}^2=\frac{1385}{12}$
4. Center of circle is $(-\frac{8}{3},\frac{37}{12})$

Q.

Find the equation of circle which touches $2x-y+3=0$ and pass through the points of intersection of the line $x+2y-1=0$ and the circle $x^2+y^2-2x+1=0$

1. $x^2+y^2+4y-1=0$

2. $x^2+y^2-4x-4y+3=0$

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

4. $x^2+y^2+4x+4y+3=0$

Q. A circle touches the hypotenuse of a right angled triangle at its middle point and passes through the mid point of the shorter side. If $a$ and $b$ $(a<b)$ be the length of the sides, then the radius of circle is

1. $\frac{b}{a}\sqrt{a^2+b^2}$ units
2. $\frac{b}{2a}\sqrt{a^2+b^2}$ units
3. $\frac{b}{4a}\sqrt{a^2+b^2}$ units
4. None of these

Q. The circle passing through $(1,-2)$ and touching the $x$-axis at $(3,0)$ also passes through the point

1. $(-2,-2)$
2. $(2,-5)$
3. $(5,-2)$
4. $(-2,5)$

Q.

Statement 1: If a line $L=0$ is tangent to the circle $S=0$, then it will also be a tangent to the circle $S+\lambda L=0$

Statement 2: If a line touches a circle, then perpendicular distance of the line from the centre of the circle is equal to the radius of the circle.

1. Statement 1 is true, statement 2 is true, statement 2 is correct explanation for statement 1

2. Statement 1 is true, Statement 2 is true, statement 2 is NOT a correct explanation for statement 1

3. Statement 1 is true, Statement 2 is false

4. Statement 1 is false, statement 2 is true