Question

Find the equation of locus of P, if the line segment joining $(4,0)$ and $(0,4)$ subtends a right angle at P.

Answer 1

Given,

$(4,0)(0,4)$

the dot product of two direction vectors must be equal to 0

$(x-4,y-0)\cdot (x-0,y-4)=0$

$x^2-4x+y^2-4y=0$

$x^2-4x+4+y^2-4y+4=0+4+4$

$(x-2{)}^2+(y-2{)}^2=8$

this is the equation of circle with center $(2,2)$ and radius $\sqrt{8}=2\sqrt{2}$