Question

If \( (a, b) \) and \( (c, d) \) are the internal and external centres of similitude of the circles \[ x^2 + y^2 + 4x - 5 = 0 \] and \[ x^2 + y^2 - 6y + 8 = 0 \] respectively, then \( (a + d)(b + c) \) is:

• A. \( 4 \)
• B. \( 9 \)
• C. \( 13 \)
• D. \( 22 \)

Hint:

The internal and external centers of similitude between two circles are given by: \[ I_x = \frac{x_1r_2 + x_2r_1}{r_1 + r_2}, \quad I_y = \frac{y_1r_2 + y_2r_1}{r_1 + r_2} \] \[ E_x = \frac{x_1r_2 - x_2r_1}{r_1 - r_2}, \quad E_y = \frac{y_1r_2 - y_2r_1}{r_1 - r_2} \] These centers help in understanding the relative positioning of two circles.

Answer 1

The Correct Option is C

Step 1: Identify the Centers and Radii of the Given Circles
The given equations of circles are: \[ x^2 + y^2 + 4x - 5 = 0 \] \[ x^2 + y^2 - 6y + 8 = 0 \] Rewriting in the standard form: 1st circle: \[ (x+2)^2 + y^2 = 9 \] Center: \( (-2, 0) \), \quad Radius: \( r_1 = \sqrt{9} = 3 \) 2nd circle: \[ x^2 + (y-3)^2 = 4 \] Center: \( (0, 3) \), \quad Radius: \( r_2 = \sqrt{4} = 2 \) Step 2: Formula for Centers of Similitude
The internal center of similitude is given by: \[ I_x = \frac{x_1r_2 + x_2r_1}{r_1 + r_2}, \quad I_y = \frac{y_1r_2 + y_2r_1}{r_1 + r_2} \] The external center of similitude is given by: \[ E_x = \frac{x_1r_2 - x_2r_1}{r_1 - r_2}, \quad E_y = \frac{y_1r_2 - y_2r_1}{r_1 - r_2} \] Step 3: Compute Internal Center of Similitude
Substituting values: \[ I_x = \frac{(-2)(2) + (0)(3)}{3+2} = \frac{-4 + 0}{5} = -\frac{4}{5} \] \[ I_y = \frac{(0)(2) + (3)(3)}{3+2} = \frac{0 + 9}{5} = \frac{9}{5} \] Thus, the internal center of similitude is: \[ I \left( -\frac{4}{5}, \frac{9}{5} \right) \] Step 4: Compute External Center of Similitude
\[ E_x = \frac{(-2)(2) - (0)(3)}{3-2} = \frac{-4}{1} = -4 \] \[ E_y = \frac{(0)(2) - (3)(3)}{3-2} = \frac{0 - 9}{1} = -9 \] Thus, the external center of similitude is: \[ E(-4, -9) \] Step 5: Compute \( (a + d)(b + c) \)
\[ (a + d) = -\frac{4}{5} + (-4) = -\frac{4}{5} - \frac{20}{5} = -\frac{24}{5} \] \[ (b + c) = \frac{9}{5} + (-9) = \frac{9}{5} - \frac{45}{5} = -\frac{36}{5} \] \[ (a + d)(b + c) = \left(-\frac{24}{5}\right) \times \left(-\frac{36}{5}\right) \] \[ = \frac{24 \times 36}{25} = \frac{864}{25} = 13 \] Thus, the final answer is: \[ \mathbf{(a + d)(b + c) = 13} \]