Question

find the equation of the circle which touches the x-axis at the origin and whose radius is 5

Answer 1

the centre of the circle will be on y-axis (on +ve side as well as on -ve side)

coz, x-axis is the tangent and centre lies on normal to the point of tangent (0,0).

C = (0,5), (0,-5)

eqn :

1) centre on +ve side
C =(0,5)
$x^{2} + {(y - 5)}^{2} = {5}^{2} \\ = > {x}^{2} + {y}^{2} - 10y = 0$

2) centre on -ve side
C =(0,-5)
${x}^{2} + {(y + 5)}^{2} = {5}^{2} \\ = > {x}^{2} + {y}^{2} + 10y = 0$

Answer 2

there can be two possible circles and hence two equations for the above conditions:-

refer to the attachment above..

hope this will help u.. :-)