Question

A line passing through the point $(4,6)$, then the locus of the mid point of the portion of line between the co-ordinate axes is

• A. $3x-2y=xy$
• B. $3x+2y=2xy$
• C. $3x+2y=xy$
• D. $3x+3y=xy$

Answer 1

The correct option is C $3x+2y=xy$

Let the equation be $\frac{x}{a}+\frac{y}{b}=1$
Since, it passes through $(4,6)$
$\Rightarrow \frac{4}{a}+\frac{6}{b}=\mathrm{1...}\left(i\right)$
The line cuts the axes at $(a,0)$ and $(0,b).$
Let $P(h.k)$ be the midpoint so
$h=\frac{a}{2},k=\frac{b}{2}\Rightarrow a=2h,b=2k$
Substituting the values of $a,b$ in equation $\left(i\right)$
$\frac{4}{2h}+\frac{6}{2k}=1$

$\Rightarrow 4k+6h=2hk$

$\Rightarrow 3h+2k=hk$

So, locus is $3x+2y=xy$