Question

A variable straight line passes through a fixed point $(a,b)$ intersecting the co-ordinates axes of $A$ and $B$. If ${}^{\prime }{O}^{\prime }$ is the origin, then the locus of the centroid of the triangle $OAB$ is

• A. $bx+ay-3xy=0$
• B. $bx+ay-2xy=0$
• C. $ax+by-3xy=0$
• D. $ax+by-2xy=0$

Answer 1

Answer: (A)

$bx+ay-3xy=0$

Let slope of line be 'm'.

Equation of line with slope m and passing through $(a,b)$ is

$(y-b)=m(x-a)$

Now, coordinate of point A$=(a-b/m,0)$

coordinate of point B $=(0,b-ma)$

Let, (h,k) be the centroid of $△AOB$.

$h=\frac{a-\frac{b}{m}}{3}$ => $m=\frac{b}{a-3h}$

$k=\frac{b-ma}{3}$

Put $m=\frac{b}{a-3h}$

=>$3k=b-\frac{ab}{a-3h}$

=> $ak+bh-3hk=0$

Now, replace $h$ with $x$ and $k$ with $y$.

=>$bx+ay-3xy=0$