Q84 / 160 · AP EAMCET 2018 (23 Apr Shift 2) · Mathematics

A variable line passing through a fixed point $(\alpha, \beta)$ intersects the coordinate axes at $A$ and $B$. If $O$ is the origin, then the locus of the centroid of the $\triangle O A B$ is

  • A
    $\beta x+\alpha y-2 \alpha \beta=0$
  • Correct answer
    B
    $\beta x+\alpha y-3 x y=0$
  • C
    $\alpha x+\beta y-\left(\alpha^2+\beta^2\right)=0$
  • D
    $\beta x+c y+3 x y=0$

Solution

Let points $A(a, 0)$ and $B(0, b)$, so equation of variable line is $$ \frac{x}{a}+\frac{y}{b}=1 $$ Since the variable line (i) passes through the point $(\alpha, \beta)$ So, $$ \frac{\alpha}{a}+\frac{\beta}{b}=1 $$ Now, centroid of $\triangle O A B$ is $\left(\frac{a}{3}, \frac{b}{3}\right)=(h, k)$ So, $$ a=3 h \text { and } b=3 k $$ From Eqs. (ii) and (iii), we are getting $$ \frac{\alpha}{3 h}+\frac{\beta}{3 k}=1 $$ On taking locus of point $(h, k)$, we are getting $$ \beta x+\alpha y-3 x y=0 . $$

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

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