Question

The equation to the line bisecting the join of $(3,-4)$ and $(5,2)$ and having its intercepts on the $x$-axis and the $y$-axis $2:1$ is

• A. $x+y-3=0$
• B. $2x-y=9$
• C. $x+2y=2$
• D. $2x+y=7$

Answer 1

The correct option is C $x+2y=2$

Let the points be $A(3,-4)$ and $B(5,2)$ and mid point of $AB=(4,-1)$.
It is given that the bisecting line intercept the co-ordinate axes in the ratio $2:1$.
$\therefore$ point of co-ordinate axes are $(2k,0)$ and $(0,k)$.

The equation of line passing through the above point is
$y-0=\frac{k-0}{0-2k}(x-2k)$
or $y=-\frac{1}{2}(x-2k)$ ..... (i)
Since, it is passing through the mid point of $AB$ i.e., $(4,-1)$
$\Rightarrow -1=-\frac{1}{2}(4-2k)$
$\Rightarrow 2=4-2k$
$\Rightarrow k=1$
Putting the value of $k$ in Eq. (i), we get
$y=-\frac{1}{2}(x-2)$
$\Rightarrow x+2y=2$