Question

If one end of a diameter of the circle \( x^2 + y^2 - 4x - 6y + 11 = 0 \) is \( (3, 4) \), then the coordinate of the other end of the diameter is:

• A. \( (1, 1) \)
• B. \( \left( \frac{1}{2}, \frac{1}{2} \right) \)
• C. \( (1, 2) \)
• D. \( (2, 1) \)
• E. \( (2, 2) \)

Hint:

The midpoint of a diameter is the center of the circle. Use this to find the missing endpoint.

Answer 1

The Correct Option is C

Rewriting the given circle equation into standard form: \[ (x - 2)^2 + (y - 3)^2 = 4 \] This shows that the center is at \( (2,3) \). 
Since the center is the midpoint of the diameter, the other endpoint is determined using the midpoint formula: \[ \left( \frac{3 + x}{2}, \frac{4 + y}{2} \right) = (2,3) \] Solving for \( x \) and \( y \), \[ \frac{3 + x}{2} = 2 \quad \Rightarrow \quad x = 1 \] \[ \frac{4 + y}{2} = 3 \quad \Rightarrow \quad y = 2 \] Thus, the other endpoint is \( (1,2) \).