Question

Let P be a variable point such that it forms a triangle of area 14 square units with two fixed points \((-3,4)\) and \((4,-3)\). Then the locus of P represents a pair of parallel lines. The distance between these two parallel lines is

• A. \(4\sqrt{2}\)
• B. \(8\)
• C. \(6\)
• D. \(3\sqrt{2}\) \bigskip

Hint:

Locus from fixed base + constant area always forms two parallel lines.

Answer 1

The Correct Option is A

Concept: Area of triangle formed by fixed base gives locus as pair of parallel lines.

Step 1: Find length of fixed side.
\[ A(-3,4),\quad B(4,-3) \] \[ AB=\sqrt{(7)^2+(-7)^2}=7\sqrt2 \]

Step 2: Use area formula.
\[ \frac12 \cdot AB \cdot h = 14 \] \[ \frac12 \cdot 7\sqrt2 \cdot h=14 \] \[ h=\frac{28}{7\sqrt2}=2\sqrt2 \]

Step 3: Distance between parallel lines.
\[ \text{Distance}=2h=4\sqrt2 \] \[ \boxed{(A)} \]