Question

What is the equation of the curve traced by point \(M\), if the sum of distances to \(A(-1, -1)\) and \(B(1, 1)\) is constant and equals \(2\sqrt{3}\)?

• A. \(2x^2 - 2xy + 2y^2 - 3 = 0\)
• B. \(2x^2 + 2xy - 2y^2 - 3 = 0\)
• C. \(2x^2 - 2xy - 2y^2 + 3 = 0\)
• D. \(2x^2 + 2xy + 2y^2 + 3 = 0\)

Answer 1

The Correct Option is A

Let \(M = (x, y)\). Then \[ \text{Given: } MA + MB = 2\sqrt{3} \Rightarrow \sqrt{(x + 1)^2 + (y + 1)^2} + \sqrt{(x - 1)^2 + (y - 1)^2} = 2\sqrt{3} \] Squaring and simplifying both sides leads to the equation of an ellipse. After algebraic simplification, it becomes: \[ 2x^2 - 2xy + 2y^2 - 3 = 0 \]