Question

If \( A(4, 0) \) and \( B(-4, 0) \) are two points, then the locus of a point \( P \) such that \( PA - PB = 4 \) is:

• A. \( 3x^2 - y^2 = 12 \)
• B. \( x^2 - 3y^2 = 12 \)
• C. \( 4(x^2 - 3y^2) = 1 \)
• D. \( 3x^2 - y^2 = 1 \)

Hint:

For problems involving the locus of points with fixed distances to two fixed points, use the distance formula and apply algebraic techniques such as squaring both sides and simplifying.

Answer 1

The Correct Option is A

We are given two points \( A(4, 0) \) and \( B(-4, 0) \), and we need to find the equation of the locus of a point \( P(x, y) \) such that the difference in distances from \( P \) to \( A \) and from \( P \) to \( B \) is 4, i.e., \( PA - PB = 4 \). Step 1: The distance from point \( P(x, y) \) to point \( A(4, 0) \) is: \[ PA = \sqrt{(x - 4)^2 + y^2} \] The distance from point \( P(x, y) \) to point \( B(-4, 0) \) is: \[ PB = \sqrt{(x + 4)^2 + y^2} \] Step 2: According to the given condition, we have: \[ PA - PB = 4 \] This gives us the equation: \[ \sqrt{(x - 4)^2 + y^2} - \sqrt{(x + 4)^2 + y^2} = 4 \] Step 3: To simplify this equation, square both sides and simplify. After simplification, we obtain the equation: \[ 3x^2 - y^2 = 12 \] Thus, the equation of the locus of point \( P \) is: \[ 3x^2 - y^2 = 12 \] % Final Answer The equation of the locus is \( 3x^2 - y^2 = 12 \).