Question

Find equation of circle which is concentric to circle $x^2+y^2-6x+7=0$ and touches the line $x+y+3=0$.

Answer 1

The two circles are concentric so they have the same centre. Centre of the given circle-

$x^2-6x+9-9+y^2+7=0$

$\Rightarrow {(x-3)}^2+{(y-0)}^2={\left(\sqrt{2}\right)}^2$

$\therefore$ Centre is $(3,0)$

Equation of the required circle is

${(x-3)}^2+y^2=a^2$

$\Rightarrow x^2+9-6x+y^2=a^2$

Whose centre is $(3,0)$ and radius is a unit.

$x+y+3=0$ is a tangent to it, so distance of line from centre $=$ its radius.

i.e. $\frac{|3+0+3|}{\sqrt{1+1}}=a$

$\Rightarrow \frac{6}{\sqrt{2}}=a\Rightarrow \frac{2\times 3}{\sqrt{2}}=a$

$\Rightarrow a=3\sqrt{2}$ units.

$\therefore$ equation of circle is ${(x-3)}^2+y^2={\left(3\sqrt{2}\right)}^2$

$\Rightarrow x^2+9-6x+y^2=18$

$\Rightarrow x^2+y^2-6x-9=0$