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A point $P$ moves such that its distances from $(1,2)$ and $(-2,3)$ are equal. Then the locus of $P$ is
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straight line
Let moving point be $\mathrm{P}(\mathrm{x}, \mathrm{y})$
$\sqrt{(\mathrm{y}-2)^{2}+(\mathrm{x}-1)^{2}}=\sqrt{(\mathrm{y}-3)^{2}+(\mathrm{x}+2)^{2}}$
$\Rightarrow(\mathrm{y}+2)^{2}+(\mathrm{x}-1)^{2}=(\mathrm{y}-3)^{2}+(\mathrm{x}+2)^{2}$
$\Rightarrow y^{2}+4+4 y+x^{2}+1-2 x=y^{2}+9-6 y+x^{2}+4 x+4$
$\Rightarrow 10 \mathrm{y}-6 \mathrm{x}-8=0$
$\therefore$ Locus of $\mathrm{P}$ is a straight line.
$\sqrt{(\mathrm{y}-2)^{2}+(\mathrm{x}-1)^{2}}=\sqrt{(\mathrm{y}-3)^{2}+(\mathrm{x}+2)^{2}}$
$\Rightarrow(\mathrm{y}+2)^{2}+(\mathrm{x}-1)^{2}=(\mathrm{y}-3)^{2}+(\mathrm{x}+2)^{2}$
$\Rightarrow y^{2}+4+4 y+x^{2}+1-2 x=y^{2}+9-6 y+x^{2}+4 x+4$
$\Rightarrow 10 \mathrm{y}-6 \mathrm{x}-8=0$
$\therefore$ Locus of $\mathrm{P}$ is a straight line.
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