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

The correct option is C A pair of parallel lines

Let the point P be $(h,k)$

Area of the triangle $ABP$ is given by

$A=\frac{1}{2}|h(3-(-2\left)\right)+5(-2-k)+3(k-3)|=9$

$\Rightarrow |5h-2k-19|=18$

Replacing $(h,K)$ by $(x,y)$, and removing the modulus, we get two pairs of equations,

$5x-2y-19=18$

$5x-2y-19=-18$

Here the lines are parallel lines as only the constant term in the equation of the lines changes.