Question

Find the equations of the circles passing through two points on Y-axis at distances 3 from the origin and having radius 5.

Answer 1

Let the required equation of the circle be

\[\left( x - h \right)^2 + \left( y - k \right)^2 = a^2\]

The circle passes through the points (0, 3) and (0, −3).

∴ \[\left( 0 - h \right)^2 + \left( 3 - k \right)^2 = a^2\] ...(1)

And,

\[\left( 0 - h \right)^2 + \left( - 3 - k \right)^2 = a^2\]...(2)

Solving (1) and (2), we get:
k=0
Given:
Radius = 5
∴ a² = 25
So, from equation (2), we have:

\[h^2 + 9 = 25 \Rightarrow h = \pm 4\]

Hence, the required equation is

\[\left( x \pm 4 \right)^2 + y^2 = 25\] , which can be rewritten as \[x^2 \pm 8x + y^2 - 9 = 0\]