Question

Two circles centred at $(2,3)$ and $(4,5)$ intersect each other. If their radii are equal, then the equation of the common chord is

• A. $x + y + 1 = 0$
• B. $x + y - 1 = 0$
• C. $x + y - 7 = 0$
• D. $x + y + 7 = 0$

Hint:

When two circles have completely identical radii, their common chord is the perpendicular bisector of the line segment joining their centers! The midpoint of $C_1(2,3)$ and $C_2(4,5)$ is $\left(\frac{2+4}{2}, \frac{3+5}{2}\right) = (3,4)$. Plug $(3,4)$ into the options: $3+4-7=0$, validating option (C) immediately!

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
We are given two circles with centers C₁(2,3) and C₂(4,5) and equal radii r, and must find the equation of their common chord.

Step 2: Key Formula or Approach:
The common chord of two intersecting circles S₁ = 0 and S₂ = 0 is given by S₁ – S₂ = 0.

Step 3: Detailed Explanation:
S₁: (x–2)²+(y–3)² = r², S₂: (x–4)²+(y–5)² = r². Subtracting: (–4x+8x)+(4–16)+(–6y+10y)+(9–25)=0 → 4x+4y–28=0 → x+y–7=0.

Step 4: Final Answer:
The common chord is x + y – 7 = 0, matching option (C).