Andhra Pradesh Board Solutions for Exercise 2: Exercise 2(b)

Find the equation and length of the common chord of the following circles: $x^2+y^2-5x-6y+4=0,x^2+y^2-2x-2=0$.

Prove that the radical axis of the circles $x^2+y^2+2gx+2fy+c=0$ and $x^2+y^2+2g^{\text{'}}x+2f^{\text{'}}y+c^{\text{'}}=0$ is the diameter of the latter circle (or the former bisects the circumference of the latter) if $2g^{\text{'}}\left(g-g^{\text{'}}\right)+2f^{\text{'}}\left(f-f^{\text{'}}\right)=c-c^{\text{'}}$.

Show that the circles $x^2+y^2+2ax+c=0$ and $x^2+y^2+2by+c=0$ touch each other if $\frac{1}{a^2}+\frac{1}{b^2}=\frac{1}{c}$.

Show that the circles $x^2+y^2-2x=0$ and $x^2+y^2+6x-6y+2=0$ touch each other. Find the coordinates of the point of contact. Is the point of contact external or internal?

Find the equation of the circle which cuts the following circles orthogonally: $x^2+y^2+4x-7=0, 2x^2+2y^2+3x+5y-9=0,x^2+y^2+y=0$.

Find the equation of the circle which cuts the following circles orthogonally: $x^2+y^2+2x+4y+1=0,2x^2+2y^2+6x+8y-3=0,x^2+y^2-2x+6y-3=0$.

Find the equation of the circle which cuts the following circles orthogonally: $x^2+y^2+2x+17y+4=0,x^2+y^2+7x+6y+11=0,x^2+y^2-x+22y+3=0$.

Find the equation of the circle which cuts the following circles orthogonally: $x^2+y^2+4x+2y+1=0,2\left(x^2+y^2\right)+8x+6y-3=0,x^2+y^2+6x-2y-3=0$.