Question

Find the radical centre of the three circles: $$ C_1: x^2 + y^2 - 1 = 0,\quad C_2: x^2 + y^2 - 8x + 15 = 0,\quad C_3: x^2 + y^2 + 10y + 24 = 0 $$

• A. \( \left(2, -\frac{5}{2} \right) \)
• B. \( \left(2, \frac{5}{2} \right) \)
• C. \( \left(-2, -\frac{5}{2} \right) \)
• D. \( \left(-2, \frac{5}{2} \right) \)

Hint:

To find the radical centre of three circles, compute two radical axes by subtracting their equations and solve for their intersection.

Answer 1

The Correct Option is A

The radical axis of two circles \( C_i \) and \( C_j \) is obtained by subtracting their equations. Step 1: \( C_2 - C_1 \): \[ (x^2 + y^2 - 8x + 15) - (x^2 + y^2 - 1) = -8x + 16 \Rightarrow \text{Radical Axis 1: } x = 2 \] Step 2: \( C_3 - C_1 \): \[ (x^2 + y^2 + 10y + 24) - (x^2 + y^2 - 1) = 10y + 25 \Rightarrow \text{Radical Axis 2: } y = -\frac{5}{2} \] Now, intersection of \( x = 2 \) and \( y = -\frac{5}{2} \) gives: \[ \boxed{ \left(2, -\frac{5}{2} \right) } \]