S L Loney Solutions for Chapter: Systems of Circles, Exercise 2: EXAMPLES XXIV

The equation of the circle, which orthogonally cuts each of the circles $x^2+y^2+2gx+c=0, x^2+y^2+2g^{\text{'}}x+c=0$ and $x^2+y^2+2hx+2ky+a=0$ is $k\left(x^2+y^2\right)+y\left(a-c\right)-ck=t$. Find $t$.

The equation of the circle which is orthogonally cutting the three circles $x^2+y^2=a^2, {\left(x-c\right)}^2+y^2=a^2$ and $x^2+{\left(y-b\right)}^2=a^2$ (where $a, b, c\ne 0$) is $p{x}^2+p{y}^2-cx-by+a^2=q$. Find $2\left(p+q+2\right)$.

The equation of the circle which is orthogonally cutting the three circles $x^2+y^2-2x+3y-7=0, x^2+y^2+5x-5y+9=0$ and $x^2+y^2+7x-9y+29=0$ is $x^2+y^2-ax+by-c=0$. Find $a+b+c$.

Show that the equation of the circle which is cutting orthogonally the circles ${\left(x-a\right)}^2+{\left(y-b\right)}^2=b^2, {\left(x-b\right)}^2+{\left(y-a\right)}^2=a^2$ and ${\left(x-a-b-c\right)}^2+y^2=ab+c^2$ is $x^2+y^2-2x\left(a+b\right)-y\left(a+b\right)+a^2+3ab+b^2=0$. (where $a\ne b$)