Question

Match the entries of Column I with Column II in the following : the vertices of a triangle are $A(a,0),B(0,b)$ and $C(a,b)$ forgiven triangle

Answer 1

Centroid is $(\frac{a+a+0}{3},0+b+b3)$ i.e. $(\frac{2a}{3},\frac{2b}{3})$

Circumcentre is mid point of diameter $AB$ i.e., $(\frac{a}{2},\frac{b}{2})$

$\therefore \left(c\right)\to \left(p\right)$
Orthocentre is the intersection of altitude $AC$ and $BC$ which intersect at $C(a,b)$

$\therefore \left(b\right)\to \left(q\right)$.
Equation of $AB$ is $\frac{x}{a}+\frac{y}{b}=1$. Its slope is $\frac{-b}{a}$, Any line perpendicular to $AB$ and passing through opposite vatex $C$ is
$y-b=\frac{a}{b}(x-a)$ or $ax-by-(a^2-b^2)=0$

$AB$ is $bx+ay-ab=0$

solving the above two, we get the foot of perpendicular from $C$ as
$\frac{x}{a{b}^2+a(a^2-b^2)}=\frac{y}{-a^{2}b+b(a^2-b^2)}=\frac{1}{a^2+b^2}$
$\therefore x=\frac{a^3}{a^2+b^2},y=\frac{b^3}{a^2+b^2}$