Question

The equation of the common chord of the circles $x^2 + y^2 - 4x - 4y = 0$ and $x^2 + y^2 - 6x - 8y + 10 = 0$ is:

• A. $x + 2y - 5 = 0$
• B. $2x + y - 5 = 0$
• C. $x - 2y + 5 = 0$
• D. $2x - y + 5 = 0$

Hint:

The common chord is always a straight line. Simply subtract the two circle equations directly to eliminate the quadratic terms $x^2$ and $y^2$.

Answer 1

The Correct Option is A

Step 1: Common chord idea.
For two circles $S_1 = 0$ and $S_2 = 0$, the common chord is simply $S_1 - S_2 = 0$. Subtracting kills the squared terms and leaves a line.

Step 2: Write the circles.
$S_1 = x^2 + y^2 - 4x - 4y$ and $S_2 = x^2 + y^2 - 6x - 8y + 10$.

Step 3: Subtract.
The $x^2$ and $y^2$ cancel: \[ (-4x + 6x) + (-4y + 8y) - 10 = 0 \]

Step 4: Collect terms.
\[ 2x + 4y - 10 = 0 \]

Step 5: Divide by 2.
\[ x + 2y - 5 = 0 \]

Step 6: Conclusion.
\[ \boxed{ x + 2y - 5 = 0 } \]