Question

Find the equations of the circles of radius 5 whose centers lie on the x-axis and pass through the point (2,3).

Answer 1

Let circles pass through the point (a,0) [This point is always on x-axis ] and the point (2,3).
As radius =5, distance between (a,0) and (2,3) is 5.
==> (a - 2)²  + (0 - 3)²  = 5²
or a² - 4a - 12 = 0 or (a - 6)(a + 2) = 0
Hence points on x-axis are (6,0) an (-2,0) which are the centres of desired circles.
Hnce equations of circle are :-
(x - 6)² + (y - 0)²  = 5² and (x + 2)² + (y - 0 )² = 5²

These are:- x² - 12x + y² +11 = 0 and x² + 4x + y² - 21 = 0