Question

find the equation of the circle which cuts the circles $x^2+y^2-9x+14=0$ and $x^2+y^2-15x+14=0$ orthogonally and passes through the piont $(2,5)$.

Answer 1

circle passes through $(2,5)$

cuts orthogonally $x^2+y^2-9x+14=0$

cuts orthoganally $x^2+y^2-15x+14=0$

let circle is $x^2+y^2+2gx+2fy+c=0$

with centre $(-9,-8)$

$4+25+4g+10f+c=0$

$4g+10f+c=-\mathrm{29.....}\left(i\right)$

$-9g=c+\mathrm{14.....}\left(ii\right)$

$-15g=c+\mathrm{14.....}\left(iii\right)$

$g=0,c=-14,B=3/2.$

circle : $x^2+y^2+37-14=0$