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

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, $\frac{a + b}{a} = \frac{ 3 +9}{2} = \frac{12}{2} = 6$