Question

A circle touches the x axis and also touches the circle with centre(0,3) and radius 2. Find the lucid of the centre of circle?

Answer 1

Dear student,

The equation of a circle touching the x-axis can be written as

${\left(x-h\right)}^2+{\left(y-k\right)}^2=k^2$

where (h,k) is the centre of the circle

For the circle with centre(0,3) and radius 2, it will cut the y-axis at (0,1) and (0,5).

As the two circles touch each other externally, the distance between the centre will be equal to the sum of the radius
(the two circles will not touch internally in this case as the first circle touches the x-axis)

So, distance between (0,3) and (h,k) is k+2

hence,

${\left(0-h\right)}^2+{\left(3-k\right)}^2={\left(k+2\right)}^2$

or

$$\begin{gathered} h^2+9+k^2-6k=4+k^2+4k \\ or{h}^2-10k+5=0 \\ Hence,thelocusofthecentreofthecircleis \\ {x}^2-10y+5=0 \\ Regards \end{gathered}$$