Key Concepts and Formulas

• Equation of a line given two points: $y - y_1 = \frac{y_2 - y_1}{x_2 - x_1}(x - x_1)$
• Equation of the chord of contact from a point $(x_1, y_1)$ to the circle $x^2 + y^2 + 2gx + 2fy + c = 0$: $xx_1 + yy_1 + g(x+x_1) + f(y+y_1) + c = 0$
• Distance from a point $(x_1, y_1)$ to a line $Ax + By + C = 0$: $d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}}$

Step-by-Step Solution

Step 1: Find the equation of line AB.

We are given points $A(4, -11)$ and $B(8, -5)$. We need to find the equation of the line passing through these two points, which will be tangent to the new circle.

The slope of the line $AB$ is: $m_{AB} = \frac{-5 - (-11)}{8 - 4} = \frac{6}{4} = \frac{3}{2}$

Using the point-slope form with point $A(4, -11)$: $y - (-11) = \frac{3}{2}(x - 4)$ $y + 11 = \frac{3}{2}x - 6$ $2y + 22 = 3x - 12$ $3x - 2y - 34 = 0 \quad (*)$ This is the equation of line $AB$.

Step 2: Find the equation of the chord of contact from point C(h, k).

The circle is given by $x^2 + y^2 - 3x + 10y - 15 = 0$. The chord of contact from point $C(h, k)$ is given by $T = 0$: $xh + yk - \frac{3(x+h)}{2} + \frac{10(y+k)}{2} - 15 = 0$ $xh + yk - \frac{3}{2}x - \frac{3}{2}h + 5y + 5k - 15 = 0$ $(h - \frac{3}{2})x + (k + 5)y - \frac{3}{2}h + 5k - 15 = 0 \quad (**)$

Step 3: Compare the two equations to find the coordinates of C(h, k).

Since both equations $(*)$ and $(**)$ represent the same line $AB$, their coefficients must be proportional: $\frac{h - \frac{3}{2}}{3} = \frac{k + 5}{-2} = \frac{-\frac{3}{2}h + 5k - 15}{-34}$

From the first two terms: $\frac{h - \frac{3}{2}}{3} = \frac{k + 5}{-2}$ $-2(h - \frac{3}{2}) = 3(k + 5)$ $-2h + 3 = 3k + 15$ $2h + 3k = -12 \quad (1)$

From the second and third terms: $\frac{k + 5}{-2} = \frac{-\frac{3}{2}h + 5k - 15}{-34}$ $17(k + 5) = -\frac{3}{2}h + 5k - 15$ $17k + 85 = -\frac{3}{2}h + 5k - 15$ $34k + 170 = -3h + 10k - 30$ $3h + 24k = -200 \quad (2)$

Multiply equation (1) by 8: $16h + 24k = -96 \quad (3)$ Subtract equation (3) from equation (2): $-13h = -104$ $h = 8$

Substitute $h = 8$ into equation (1): $2(8) + 3k = -12$ $16 + 3k = -12$ $3k = -28$ $k = -\frac{28}{3}$

Thus, the coordinates of point $C$ are $(8, -\frac{28}{3})$.

Step 4: Calculate the radius of the new circle.

The radius of the new circle is the perpendicular distance from $C(8, -\frac{28}{3})$ to the line $3x - 2y - 34 = 0$. $r = \frac{|3(8) - 2(-\frac{28}{3}) - 34|}{\sqrt{3^2 + (-2)^2}}$ $r = \frac{|24 + \frac{56}{3} - 34|}{\sqrt{13}}$ $r = \frac{|\frac{-30 + 56}{3}|}{\sqrt{13}}$ $r = \frac{\frac{26}{3}}{\sqrt{13}} = \frac{26}{3\sqrt{13}} = \frac{26\sqrt{13}}{3(13)} = \frac{2\sqrt{13}}{3}$

Common Mistakes & Tips

• Be careful with the signs when applying the distance formula and the chord of contact formula.
• Simplify equations to their standard forms as much as possible to avoid errors.
• When comparing coefficients, ensure that the equations represent the same line.

Summary

We found the equation of the line $AB$, then the equation of the chord of contact from $C(h,k)$ to the given circle. By comparing the coefficients of these two equations, we found the coordinates of $C$. Finally, we calculated the radius of the new circle as the distance from $C$ to the line $AB$, which is $\frac{2\sqrt{13}}{3}$.

Final Answer

The final answer is $\boxed{\frac{2\sqrt{13}}{3}}$, which corresponds to option (A).