Question

If $A =(5,3) , B = (3, -2)$ and a point $P$ is such that the area of the triangle $PAB$ is $9$, then the locus of $P$ represents

• A. a circle
• B. a pair of coincident lines
• C. a pair of parallel lines
• D. a pair of perpendicular lines

Answer 1

Answer: (C)

a pair of parallel lines

Answer 2

Let $P(x, y), A=(5,3), B=(3,-2)$ Area of $\Delta P A B=\frac{1}{2} \begin{vmatrix}x & y & 1 \\\ 5 & 3 & 1 \\\ 3 & -2 & 1\end{vmatrix}$ $\Rightarrow 9=\frac{1}{2}|x(3+2)-y(5-3)+(-10-9)|$ $\Rightarrow 18=|5\,x-2\, y-19|$ $\Rightarrow 5 \,x-2 \,y-19=|18|$ $\Rightarrow 5 \,x-2 \,y-19=\pm 18$ $\Rightarrow 5 \,x-2 \,y=19 \pm 18$ $\Rightarrow 5 \,x-2 \,y=1$ or $5 \,x-2 \,y=37$ which represents a pair of parallel lines.