Equation of the circle(s) touching the x-axis at a distance 3 from the origin and having an intercept of length $2\sqrt7$ on the y-axis

Question

Find the equation of the circle(s) touching the x-axis at a distance 3 from the origin and having an intercept of length $2\sqrt7$ on the y-axis, is...

I want specifically a geometrical method of solving this question.

This is what I did:

As you can see, from the diagram that I have assumed that the centre is $(3,y_1)$. For the points, the circle intercepts the y-axis, x=0.

So the questions that want answers for are:

1. Is the assumption I have made correct? (Just a bit unsure about it.)
2. How do I find the value of $y_1$?
3. How to find the length of the radius?

Answer 1

Actually, there are 4 such circles which satisfy the condition. Ponder:

Let's focus our attention to only the circle in the first quadrant.

I have hidden the numbers on the axes so that we don't know the value of $y_1$

Look at the following diagram:

Convince yourself with the following facts

• $BC=\frac{1}{2}BD=\sqrt{7}$
• $AB=AE=r$
• $AC=3$
• $AB^{2}=AC^{2}+BC^{2}$
• $r=4$
• $y_{1}=4$

Similarly, by symmetry you can find the parameters for the other circles.

Answer 2

Another approach:

We translate the center of circle to point $(3, \sqrt 7)$, the equation of circle will be:

$$(x-3)^2+(y-\sqrt 7)^2=R^2$$

In this position the circle passes the origin, so we must have:

$$(0-3)^2+(0-\sqrt 7)=R^2$$

Which gives $R=4$ and $y_1=4$