Question:

Step 1: Determine the coordinates of the chord endpoints. Let the endpoints of the chord be $A=(x_1, y_1)$ and $B=(x_2, y_2)$. The chord is trisected at the points $P_1 = \left(\frac{1}{3}, \frac{1}{3}\right)$ and $P_2 = \left(\frac{8}{3}, \frac{8}{3}\right)$. Using the section formula, $P_1$ divides the segment $AB$ in the ratio $1:2$, and $P_2$ divides $AB$ in the ratio $2:1$. For $P_1$: $$ \left(\frac{2x_1 + x_2}{3}, \frac{2y_1 + y_2}{3}\right) = \left(\frac{1}{3}, \frac{1}{3}\right) $$ This yields the equations: $$ 2x_1 + x_2 = 1 \quad (1) $$ $$ 2y_1 + y_2 = 1 \quad (2) $$ For $P_2$: $$ \left(\frac{x_1 + 2x_2}{3}, \frac{y_1 + 2y_2}{3}\right) = \left(\frac{8}{3}, \frac{8}{3}\right) $$ This yields the equations: $$ x_1 + 2x_2 = 8 \quad (3) $$ $$ y_1 + 2y_2 = 8 \quad (4) $$ Adding equations (1) and (3): $$ (2x_1 + x_2) + (x_1 + 2x_2) = 1 + 8 $$ $$ 3x_1 + 3x_2 = 9 \implies x_1 + x_2 = 3 \quad (5) $$ Now, solve the system of equations for $x_1$ and $x_2$: $$ 2x_1 + x_2 = 1 $$ $$ x_1 + x_2 = 3 $$ Subtracting the second equation from the first gives $x_1 = 1 - 3 = -2$. Substituting $x_1 = -2$ into $x_1 + x_2 = 3$ gives $-2 + x_2 = 3 \implies x_2 = 5$. Similarly, for $y_1$ and $y_2$, adding equations (2) and (4): $$ (2y_1 + y_2) + (y_1 + 2y_2) = 1 + 8 $$ $$ 3y_1 + 3y_2 = 9 \implies y_1 + y_2 = 3 \quad (6) $$ Solving the system for $y_1$ and $y_2$: $$ 2y_1 + y_2 = 1 $$ $$ y_1 + y_2 = 3 $$ Subtracting the second equation from the first gives $y_1 = 1 - 3 = -2$. Substituting $y_1 = -2$ into $y_1 + y_2 = 3$ gives $-2 + y_2 = 3 \implies y_2 = 5$. Thus, the endpoints of the chord are $A=(-2, -2)$ and $B=(5, 5)$. Step 2: Determine the value of $c$ for the circle. The equation of the circle is $x^2 + y^2 - 4x - 2y - c = 0$. Since the endpoints of the chord lie on the circle, substitute the coordinates of $A=(-2, -2)$ into the circle equation: $$ (-2)^2 + (-2)^2 - 4(-2) - 2(-2) - c = 0 $$ $$ 4 + 4 + 8 + 4 - c = 0 $$ $$ 20 - c = 0 \implies c = 20 $$ The equation of the circle is therefore $x^2 + y^2 - 4x - 2y - 20 = 0$. Step 3: Calculate the radius of the circle. To find the radius, convert the circle equation to its standard form $(x-h)^2 + (y-k)^2 = r^2$ by completing the square: $$ (x^2 - 4x) + (y^2 - 2y) = 20 $$ $$ (x^2 - 4x + 4) + (y^2 - 2y + 1) = 20 + 4 + 1 $$ $$ (x - 2)^2 + (y - 1)^2 = 25 $$ The radius $r$ is the square root of the constant term on the right side: $$ r = \sqrt{25} = 5 $$ The radius of the circle is 5.