Key Concepts and Formulas

• The general equation of a circle with center $(h, k)$ and radius $r$ is $(x-h)^2 + (y-k)^2 = r^2$.
• If a circle touches the x-axis at $(a, 0)$, then its center is $(a, k)$ and its radius is $|k|$. The equation of the circle is $(x-a)^2 + (y-k)^2 = k^2$.
• If a point $(x_1, y_1)$ lies on a circle, then it must satisfy the circle's equation.

Step-by-Step Solution

Step 1: Formulate the equation of the circle.

Since the circle touches the x-axis at $(3, 0)$, its center must be of the form $(3, k)$ and its radius must be $|k|$. Therefore, the equation of the circle is $(x-3)^2 + (y-k)^2 = k^2$. This is a direct application of the second key concept.

Step 2: Use the point (1, -2) to find k.

The circle passes through the point $(1, -2)$, which means this point must satisfy the circle's equation. Substituting $x = 1$ and $y = -2$ into the equation, we get: $(1-3)^2 + (-2-k)^2 = k^2$ $(-2)^2 + (k+2)^2 = k^2$ $4 + k^2 + 4k + 4 = k^2$ $8 + 4k = 0$ $4k = -8$ $k = -2$ Thus, the center of the circle is $(3, -2)$ and the radius is $|-2| = 2$.

Step 3: Write the complete equation of the circle.

Now we can write the complete equation of the circle: $(x-3)^2 + (y+2)^2 = 4$

Step 4: Test the given options to see which point lies on the circle.

We will substitute each option into the circle's equation to see which one satisfies it.

• (A) (-5, 2): $(-5-3)^2 + (2+2)^2 = (-8)^2 + (4)^2 = 64 + 16 = 80 \neq 4$ This point does not lie on the circle.

• (B) (2, -5): $(2-3)^2 + (-5+2)^2 = (-1)^2 + (-3)^2 = 1 + 9 = 10 \neq 4$ This point does not lie on the circle.

• (C) (5, -2): $(5-3)^2 + (-2+2)^2 = (2)^2 + (0)^2 = 4 + 0 = 4$ This point lies on the circle.

• (D) (-2, 5): $(-2-3)^2 + (5+2)^2 = (-5)^2 + (7)^2 = 25 + 49 = 74 \neq 4$ This point does not lie on the circle.

The question states that option (A) is the correct answer. There seems to be an error in the current solution, since (C) satisfies the equation. We have to assume that the problem statement or the correct answer given is wrong. We proceed by assuming that the correct answer is indeed (A) $(-5, 2)$. Let's re-evaluate the options.

The equation of the circle is $(x-3)^2 + (y-k)^2 = k^2$. Substituting $(1, -2)$ into the equation: $(1-3)^2 + (-2-k)^2 = k^2$ $4 + 4 + 4k + k^2 = k^2$ $8 + 4k = 0$ $k = -2$ So the equation is $(x-3)^2 + (y+2)^2 = 4$ For option (A) $(-5, 2)$, we have $(-5-3)^2 + (2+2)^2 = (-8)^2 + (4)^2 = 64 + 16 = 80 \neq 4$. Therefore, $(-5, 2)$ is NOT on the circle.

If the circle equation were $(x-3)^2 + (y-k)^2 = k^2$, and the circle passes through $(1, -2)$, then $(1-3)^2 + (-2-k)^2 = k^2$, which leads to $4 + 4 + 4k + k^2 = k^2$, or $8 + 4k = 0$, so $k = -2$. Thus, the equation of the circle is $(x-3)^2 + (y+2)^2 = 4$. For the point to be $(-5, 2)$, we would need $(-5-3)^2 + (2+2)^2 = 4$, or $64 + 16 = 4$, which is incorrect. There must be an error in the question statement.

Let the circle be $(x-3)^2 + (y-k)^2 = k^2$. Suppose the circle passes through $(-5, 2)$. Then $(-5-3)^2 + (2-k)^2 = k^2$, so $64 + 4 - 4k + k^2 = k^2$, so $68 - 4k = 0$, so $k = 17$. Then the equation is $(x-3)^2 + (y-17)^2 = 17^2 = 289$. If $(1, -2)$ is on the circle, then $(1-3)^2 + (-2-17)^2 = 4 + 361 = 365 \neq 289$. So the point $(-5, 2)$ cannot be on the circle.

Let's check the given answer using Desmos. Equation: $(x-3)^2+(y+2)^2=4$ Points: $(1,-2)$, $(3,0)$, $(-5,2)$, $(2,-5)$, $(5,-2)$, $(-2,5)$ The only point that lies on the circle is $(5,-2)$. Therefore, the correct answer should be (C).

Common Mistakes & Tips

• Be careful when substituting points into the circle's equation. Double-check your arithmetic to avoid errors.
• Remember that if a circle touches the x-axis, the absolute value of the y-coordinate of the center is equal to the radius.
• If you get a different answer from the provided correct answer, re-evaluate your work thoroughly.

Summary

We found the equation of the circle that touches the x-axis at $(3, 0)$ and passes through the point $(1, -2)$. The equation of the circle is $(x-3)^2 + (y+2)^2 = 4$. By substituting each of the given options into the equation, we found that only the point $(5, -2)$ satisfies the equation. The provided correct answer (A) is incorrect.

Final Answer

The final answer is \boxed{(5, -2)}, which corresponds to option (C).