If a circle passes through the point (a,b) and cuts the circle x^2+y^2=4 orthogonally, the…

Question

If a circle passes through the point $(a,b)$ and cuts the circle $x^2+y^2=4$ orthogonally, then the locus of its centre is:

Accepted Answer

Let the variable circle is

$${x^2} + {y^2} + 2gx + 2fy + c = 0\,\,\,\,\,\,\,\,...\left( 1 \right)$$

It passes through $(a,b)$

$$\therefore$$ $${a^2} + {b^2} + 2ga + 2fb + c = 0\,\,\,\,\,\,\,...\left( 2 \right)$$

$(1)$ cuts ${x^2} + {y^2} = 4$ orthogonally

$$\therefore$$ $$2\left( {g \times 0 + f \times 0} \right) = c - 4 \Rightarrow c = 4$$

$$\therefore$$  from $(2)$ $$\,\,\,{a^2} + {b^2} + 2ga + 2fb + 4 = 0$$

$$\therefore$$ Locus of center $$\left( { - g, - f} \right)$$ is

${a^2} + {b^2} - 2ax - 2by + 4 = 0$

or $2ax + 2by = {a^2} + {b^2} + 4$

Solutions · 1