Question

The coordinates of the three points $O$, $A$ and $B$ are $(0,0),$ $(0,4)$ and $(6,0)$ respectively. If a point $P$ moves so that the area of $\Delta POA$ is always twice the area of $\Delta POB$, the locus of $P$ is

• A. $x-3y=0$
• B. $x+3y=0$
• C. $3x+4y=0$
• D. $3x-4y=0$

Answer 1

Answer: (A), (B)

$x-3y=0$
$x+3y=0$

Let the coordinates of point $P$ be $(x,y)$
$\therefore \Delta POA=\left|\left|\begin{array}{ccc}x& y& 1\\ 0& 0& 1\\ 0& 4& 1\end{array}\right|\right|=|-4x|=|4x|$, (expanding along first column)
And $\Delta POB=\left|\left|\begin{array}{ccc}x& y& 1\\ 0& 0& 1\\ 6& 0& 1\end{array}\right|\right|=|6y|$, (expanding along second column)
Now given $\Delta POA=2\cdot \Delta POB$
$\Rightarrow |4x|=2\cdot |6y|\Rightarrow |x|=3|y|$
Squaring both sides we get, $x^2=9y^2\Rightarrow x^2-9y^2=0$
Hence required locus of $P$ is $x^2-9y^2=0$
i.e. $x+3y=0$ or $x-3y=0$