Animated Solution for Mathematics - Circles: Circle(s) touching x-axis at a distance 3 from the origin and having an intercept of length $2\sqrt{7}$ on y-axis is (are)

We are given that a circle is tangent to the $x$-axis at the point $(3,0)$. Because the tangent is horizontal, the radius at this point must be vertical.

This implies that the center of the circle, $C$, must lie on the vertical line $x=3$. Let the center be $C(3,k)$.

Since the circle touches the $x$-axis, the distance from the center $(3,k)$ to the $x$-axis is equal to the radius $r$. Therefore, we have the relationship:

The circle cuts an intercept of length $2\sqrt{7}$ on the $y$-axis. The perpendicular distance from the center $(3,k)$ to the $y$-axis is the $x$-coordinate of the center, which is $3$.

Consider the right-angled triangle formed by the center of the circle, the origin, and the midpoint of the chord on the $y$-axis. The base of this triangle is $3$, and the height is half the length of the chord, which is $\sqrt{7}$.

By the Pythagorean theorem, the hypotenuse of this triangle is the radius $r$. Thus:

Given $r^2=16$ and $r=|k|$, we find that $k^2=16$, which yields $k=\pm 4$. This results in two possible centers: $(3,4)$ and $(3,-4)$, both with a radius of $r=4$.

For the center $(3,4)$, the equation is:

Expanding this expression:

For the center $(3,-4)$, the equation is:

Expanding this expression:

The two possible equations for the circle are $x^2+y^2-6x-8y+9=0$ and $x^2+y^2-6x+8y+9=0$.