If the ends of the hypotenuse of a right angled triangle are (0, a) and (a, 0) then the locus of the third vertex is 

Let $A=\left(a,o\right),B=\left(o,a\right)\&P=\left(x,y\right)$
$\Delta ABP$ is a right angle triangle
$\Rightarrow$ Locus of ‘P’ is a circle AB as ends of a diameter
$\left(x-a\right)\left(x-o\right)+\left(y-o\right)\left(y-a\right)=0$
$x^2+y^2-ax-ay=0$