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If $A=(-1,2)$ and $B=(1,-2)$ are two points and $P$ is a variable point such that the area of $\triangle P A B$ is always one, then the equation of the locus of $P$ is
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Verified Answer
The correct answer is:
$4 x^2+4 x y+y^2=1$
Let $P$ be $(x, y)$.
We have, ar $(\triangle P A B)=1$
$$
\begin{array}{rlrlrl}
& \left|\begin{array}{ccc}
x & y & 1 \\
-1 & 2 & 1 \\
1 & -2 & 1
\end{array}\right| = \pm 1 \\
& x(2+2)-y(-1-1)+1(2-2) = \pm 2 \\
\Rightarrow & 4 x+2 y = \pm 2 \\
\Rightarrow & 2 x+y = \pm 1 \\
\Rightarrow & 4 x^2+4 x y+y^2 =1
\end{array}
$$
We have, ar $(\triangle P A B)=1$
$$
\begin{array}{rlrlrl}
& \left|\begin{array}{ccc}
x & y & 1 \\
-1 & 2 & 1 \\
1 & -2 & 1
\end{array}\right| = \pm 1 \\
& x(2+2)-y(-1-1)+1(2-2) = \pm 2 \\
\Rightarrow & 4 x+2 y = \pm 2 \\
\Rightarrow & 2 x+y = \pm 1 \\
\Rightarrow & 4 x^2+4 x y+y^2 =1
\end{array}
$$
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