Question

Let $A(2,-3)$ and $B(-2,3)$ be vertices of a triangle $ABC$. If the centroid of this triangle moves on the line $2x+3y=1$. If the locus of the vertex $C$ is the line $ax+by=c$, then $\frac{b+c}{a}$ is

Answer 1

Answer: 6

Let the vertex $C$ be $(h,k)$, then the
centroid ot $\Delta ABC$ is $\left(\frac{2-2+h}{3},\frac{-3+1+k}{3}\right)$
or $\left(\frac{h}{3},\frac{-2+k}{3}\right)$.It lies on $2x+3y=1$
$\Rightarrow \frac{2h}{3}-2+k=1$
$\Rightarrow 2h+3k=9$
$\Rightarrow$ Locus of $C$ is $2x+3y=9$
But, locus is the line $ax+by=c$
Then, $a=2,b=3,c=9$
Hence, $\frac{a+b}{a}=\frac{3+9}{2}=\frac{12}{2}=6$