Question

If P(-1, 1) and Q(2, 3) are two fixed points, find the locus of a point A such that the area of $\Delta$ APQ = 8 sq. units.

• A. 2x-3y=21
• B. 2x-3y=-21
• C. 2x-3y=-11
• D. 2x-3y=11

Answer 1

Answer: (B), (D)

The correct options are
B 2x-3y=-21
D 2x-3y=11

Let the coordinates of point A be (x, y).
Given ar($\Delta$ APQ) = 8 sq. units.
$\Rightarrow \frac{1}{2}\mathrm{mod}\left(\right(\begin{array}{ccc}-1& 1& 1\\ 2& 3& 1\\ x& y& 1\end{array}\left|\right)=8,\mathrm{where}\mathrm{mod}(\cdot )\mathrm{denotes}\mathrm{modulus}.\left(\mathrm{Since}\mathrm{area}\mathrm{can}\text{'}t\mathrm{be}\mathrm{negative}\right]\Rightarrow \mathrm{mod}\left(-\left(3-y\right)-\left(2-x\right)+\left(2y-3x\right)\right)=16\Rightarrow \mathrm{mod}\left(-2x+3y-5\right)=16\Rightarrow -2x+3y-5=16\mathrm{or}-2x+3y-5=-16\Rightarrow 2x-3y=-21\mathrm{or}2x-3y=11$