Question

Find the equation of the circle which passes through two points on the x- axis which are at distances $4$ units from the origin and whose radius is $5$ units.

Answer 1

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There are two circles which pass through two points A and A' on the x-axis which are at a distance of 4 units from the origin.

The center of there circles lie on the y-axis

In $\Delta OAC,A{C}^2=O{A}^2+O{C}^2$

$\Rightarrow 5^2=4^2+O{C}^2\Rightarrow O{C}^2=5^2-4^2=25-16=9$

$\therefore OC=3$ units

So, the coordinates of the centers of the required circles are C(0,3) and C'(0,-3)

Hence, the equation of the required circles are $(x-0{)}^2+(y\pm 3{)}^2=5^2$

$\Rightarrow x^2+y^2\pm 6y-16=0$