Question

Find the equation of the locus of a point that forms a triangle of area 5 units with the points A(2,3) and B(-1,4).

Answer 1

Let $P(x_1,y_1)$ be point on the locus, $(x_2,y_2)=(2,3)$ and $(x_3,y_3)=(-1,4)$

Given, area of $△PAB$ = 5 sq. units

Therefore,

$\frac{1}{2}\left|\begin{array}{cc}{x}_1-2& 2-(-1)\\ y_1-3& 3-4\end{array}\right|=5$

$\left|\begin{array}{cc}{x}_1-2& 3\\ y_1-3& -1\end{array}\right|=10$

$-(x_1-2)-3(y_1-3)=\pm 10$

$x_1+3y_1+11=\pm 10$

Required equation is $x+3y=10-11$ or $x+3y=-10-11$

That is,

$x+3y+1=0$ or $x+3y+21=0$.