A line cuts x-axis at A and y-axis at B such that $AB = l$ Match the following

Column-I and Column II (Locus of the point lies on)

$\text{A) Let }\mathrm{A}=(a,0),\mathrm{B}=(0,b)\Rightarrow a^2+b^2=l^2$

If the circumcentre is (x, y)

$x=\frac{a}{2},y=\frac{b}{2}\Rightarrow x^2+y^2=\frac{a^2+b^2}{4}=\frac{l^2}{4}$

B) If orthocentre is the point O, then locus contains just one point i..e, $x^2+y^2=0$

C) Let (x, y) be incentre $x=\frac{ab}{a+b+l},y=\frac{ab}{a+b+l}\Rightarrow x=y$

D) Let (x,y) be centroid $\Rightarrow x=\frac{a}{3},y=\frac{b}{3}\Rightarrow x^2+y^2=\frac{l^2}{9}$