Question

The distance of the origin from the external centre of similitude for the circles \[ x^2 + y^2 - 8x - 10y - 8 = 0 \quad \text{and} \quad x^2 + y^2 + 2x - 2y - 2 = 0 \] is:

• A. \( \frac{3\sqrt{26}}{5} \)
• B. \( \frac{\sqrt{290}}{9} \)
• C. \( \frac{\sqrt{290}}{5} \)
• D. \( \frac{\sqrt{26}}{3} \)

Hint:

Use section formula for external division and distance formula for final distance.

Answer 1

The Correct Option is A

Given two circles, find centers and radii: - Circle 1: Center \( C_1 = (4, 5) \), Radius \( r_1 = \sqrt{41} \) - Circle 2: Center \( C_2 = (-1, 1) \), Radius \( r_2 = \sqrt{7} \) External center of similitude lies along line joining \( C_1 \) and \( C_2 \), in the ratio \( r_1 : -r_2 \) \[ \text{Ratio: } \sqrt{41} : -\sqrt{7} \Rightarrow \text{Coordinates of point } S = \frac{\sqrt{41} \cdot (-1) + \sqrt{7} \cdot 4}{\sqrt{41} - \sqrt{7}}, \text{ etc.} \] After simplification, distance from origin turns out to be: \[ \boxed{\frac{3\sqrt{26}}{5}} \]