Q56 / 155 · WBJEE 2017 · Mathematics

Let $A(2,-3)$ and $B(-2,1)$ be two angular points of $\Delta A B C$. If the centroid of the triangle moves on the line $2 x+3 y=1$, then the locus of the angular point $C$ is given by

  • Correct answer
    A
    $2 x+3 y=9$
  • B
    $2 x-3 y=9$
  • C
    $3 x+2 y=5$
  • D
    $3 x-2 y=3$

Solution

Let the coordinates of $C$ be $(\alpha, \beta)$. $\therefore$ Coordinates of centroid $$ \begin{array}{l} =\left(\frac{2-2+\alpha}{3}, \frac{-3+1+\beta}{3}\right) \\ =\left(\frac{\alpha}{3}, \frac{\beta-2}{3}\right) \end{array} $$ Since, centroid lie on $2 x+3 y=1$ $\therefore \quad \frac{2 \alpha}{3}+3\left(\frac{\beta-2}{3}\right)=1$ $\Rightarrow \quad \frac{2 \alpha}{3}+\frac{3 \beta-6}{3}=1$ $\Rightarrow \quad 2 \alpha+3 \beta-6=3 \Rightarrow 2 \alpha+3 \beta=9$ $\therefore$ Locus of point $C$ will be $2 x+3 y=9$

Hence, the correct answer is option: 'A'.

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