The problem asks to find the equation of a circle with radius 5, where the center of the circle lies on the x-axis and the circle passes through the point (2, 3).

Step-by-Step Solution:

1. General equation of a circle: The general form of the equation of a circle is: $(x - h)^2 + (y - k)^2 = r^2$ where $(h, k)$ is the center of the circle and $r$ is the radius.

2. Given information:

   • The radius of the circle is $5$, so $r = 5$.
   • The center of the circle lies on the x-axis, meaning $k = 0$. Therefore, the center is at $(h, 0)$.
   • The circle passes through the point $(2, 3)$.

3. Substitute into the circle equation: We substitute the given values into the circle equation: $(x - h)^2 + (y - 0)^2 = 5^2$ which simplifies to: $(x - h)^2 + y^2 = 25$

4. Use the point (2, 3): Since the point $(2, 3)$ lies on the circle, we substitute $x = 2$ and $y = 3$ into the equation: $(2 - h)^2 + 3^2 = 25$ Simplifying: $(2 - h)^2 + 9 = 25$ $(2 - h)^2 = 16$

5. Solve for $h$: Taking the square root of both sides: $2 - h = \pm 4$ This gives two possible values for $h$: $h = 2 - 4 = -2 \quad \text{or} \quad h = 2 + 4 = 6$

6. Equations of the circle: Thus, the center of the circle can be either $(-2, 0)$ or $(6, 0)$. Therefore, we have two possible equations for the circle:

   For $h = -2$: $(x + 2)^2 + y^2 = 25$

   For $h = 6$: $(x - 6)^2 + y^2 = 25$

So, the equation of the circle can be either $(x + 2)^2 + y^2 = 25$ or $(x - 6)^2 + y^2 = 25$.

Do you want any further details or have any questions? Here are 5 related questions to explore:

1. How do we determine the center of a circle if it's on the y-axis instead of the x-axis?
2. Can you find the distance between the two possible centers of the circle?
3. What happens if the radius changes to a different value while keeping the same center?
4. How can we find the tangent line to this circle at the point (2, 3)?
5. How can we derive the equation of a circle if it's inscribed in a square?

Tip: To verify whether a point lies on a circle, simply substitute the coordinates of the point into the circle's equation.