Question

If the line segment joining $(2,-3)$ and $(5,6)$ is divided by the $y-$axis, then

• A. the point of division is $(0,-9)$
• B. the point of division is $(0,9)$
• C. the ratio is $2:5$ externally
• D. the ratio is $2:5$ internally

Answer 1

Answer: (A), (C)

The correct options are
A the point of division is $(0,-9)$
C the ratio is $2:5$ externally

Let $A\equiv (2,-3)$ and $B\equiv (5,6).$
Let $y-$axis divides the line segment $AB$ internally in the ratio $m:n$ at point $H.$

Then, $H\equiv (\frac{5m+2n}{m+n},\frac{6m-3n}{m+n})$

Since $H$ lies on $y-$axis, therefore, $x-$coordinate of $H$ will be zero.
$\therefore \frac{5m+2n}{m+n}=0$
$\therefore 5m=-2n$
$\Rightarrow \frac{m}{n}=\frac{-2}{5}$
Hence, $y-$axis divides the line segment $AB$ externally.
$\therefore H\equiv (0,\frac{2\times 6-5(-3)}{2-5})=(0,-9)$