Key Concepts and Formulas

• The perpendicular distance, $d$, from a point $(x_1, y_1)$ to a line $Ax + By + C = 0$ is given by: $d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}}$
• If a chord of length $L$ is at a distance $d$ from the center of a circle with radius $r$, then: $r^2 = d^2 + \left(\frac{L}{2}\right)^2$

Step-by-Step Solution

Step 1: Find the distance from the center of the circle to the line.

The equation of the circle is $x^2 + y^2 = r^2$, so the center of the circle is $(0, 0)$. The equation of the line is $y - 2x = 3$, which can be rewritten as $2x - y + 3 = 0$. We want to find the perpendicular distance, $d$, from the point $(0, 0)$ to the line $2x - y + 3 = 0$. Using the formula for the distance from a point to a line: $d = \frac{|2(0) - (0) + 3|}{\sqrt{2^2 + (-1)^2}} = \frac{|3|}{\sqrt{4 + 1}} = \frac{3}{\sqrt{5}}$

Step 2: Use the relationship between the radius, chord length, and distance to the center.

The length of the chord is given as $r$. We know that the radius of the circle is $r$, and the distance from the center to the chord is $d = \frac{3}{\sqrt{5}}$. Applying the Pythagorean theorem to the right triangle formed by the radius, half the chord length, and the perpendicular distance: $r^2 = d^2 + \left(\frac{r}{2}\right)^2$ Substitute the value of $d$: $r^2 = \left(\frac{3}{\sqrt{5}}\right)^2 + \left(\frac{r}{2}\right)^2$ $r^2 = \frac{9}{5} + \frac{r^2}{4}$

Step 3: Solve for $r^2$.

Subtract $\frac{r^2}{4}$ from both sides: $r^2 - \frac{r^2}{4} = \frac{9}{5}$ $\frac{3r^2}{4} = \frac{9}{5}$ Multiply both sides by $\frac{4}{3}$: $r^2 = \frac{9}{5} \cdot \frac{4}{3} = \frac{36}{15} = \frac{12}{5}$ Multiplying both sides by $\frac{4}{3}$ gives us $r^2 = \frac{9}{5} \cdot \frac{4}{3} = \frac{3}{5} \cdot 4 = \frac{12}{5}$ There seems to be an error in the provided correct answer. The correct value of $r^2$ is $\frac{12}{5}$.

Using the Pythagorean theorem: $r^2 = d^2 + \left(\frac{L}{2}\right)^2$ Given that $L = r$, $r^2 = \left(\frac{3}{\sqrt{5}}\right)^2 + \left(\frac{r}{2}\right)^2 = \frac{9}{5} + \frac{r^2}{4}$ $r^2 - \frac{r^2}{4} = \frac{9}{5}$ $\frac{3r^2}{4} = \frac{9}{5}$ $r^2 = \frac{9}{5} \cdot \frac{4}{3} = \frac{3}{5} \cdot 4 = \frac{12}{5}$

Common Mistakes & Tips

• Be careful with the formula for the distance from a point to a line. Make sure you have the correct coefficients and the correct sign.
• Remember to halve the chord length before squaring it in the Pythagorean theorem.
• Double-check your algebraic manipulations to avoid errors when solving for $r^2$.

Summary

We found the perpendicular distance from the center of the circle to the given line. Then, we used the relationship between the radius, chord length, and perpendicular distance to set up an equation. Finally, we solved the equation for $r^2$, obtaining $r^2 = \frac{12}{5}$. The provided answer of $\frac{9}{5}$ is incorrect.

Final Answer

The final answer is $\boxed{\frac{12}{5}}$, which corresponds to option (C).