Question

Find equations to the circles touching y-axis at $(0,3)$ and making intercept of $8$ units on the X-axis

Answer 1

Equation of circle touches $y-$axis at $(0,3)$ is

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

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

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

Since the length of intercept on $x-$axis at this circle is $8$ unit.

$2\sqrt{a^2-9}=8$

$\Rightarrow \sqrt{a^2-9}=4$

$\Rightarrow a^2-9=16$

$\Rightarrow a^2=16+9=25$

$\Rightarrow a=\sqrt{25}=5$ since $a>0$

The required equation is $x^2+y^2-2\times 5x-6y+9=0$

or $x^2+y^2-10x-6y+9=0$