Question

Find the equation of circle which touches both the negative axes and has its center on the line `x-2y=3`.?

Answer 1

`(x+3)^2+(y+3)^2=9`

Explanation:

The center of the circle must have equal `x` and `y` coordinate values if the circle touches both axes.

For the line `x-2y=3`
this means that (after substituting `x` for `y`, since they need to be equal)
`color(white)("XXX")x-2x=3color(white)("xxx")rarrcolor(white)("xxx")x=-3`
and similarly
`color(white)("XXX")y=-3`

With a center at `(color(green)(-3),color(blue)(-3))` and a radius of `color(magenta)3` (since this is the distance from the center to both axes),
the equation of the circle is
`color(white)("XXX")(x-(color(green)(-3)))^2+(y-(color(blue)(-3)))^2=color(red)3^2`
or
`color(white)("XXX")(x+3)^2+(y+3)^2=9`