Question

Find the equation of the locus of point P ,if the point A(2,3) and B(1,4) are at equal distance at P.

Answer 1

Given: Condition that the point $A(2,3)$ and $B(1,4)$ are at equal distance at $P(x,y)$.

$\Rightarrow AP=PB$

Using distance formula in the above condition, we have

$\sqrt{{(x-2)}^2+{(y-3)}^2}=\sqrt{{(x-1)}^2+{(y-4)}^2}$

Squaring both sides, we get

${(x-2)}^2+{(y-3)}^2={(x-1)}^2+{(y-4)}^2$

$\Rightarrow x^2+4-4x+y^2-6y+9=x^2+1-2x+y^2-8y+16$

$\Rightarrow 4-4x-6y+9-1+2x+8y-16=0$

$\Rightarrow -2x+2y-4=0$

or $x-y+2=0$ is the equation of the locus.