Question

In what ratio is the line segment joining the points (−2, −3) and (3, 7) divided by the y-axis? Also, find the coordinates of the point of division.

Answer 1

Let the y - axis cut the join A(-2, -3) and B(3, 7) at the point P in the ratio k : 1.
Then, by section formula, the co-ordinates of P are
$\left(\frac{3k-2}{k+1},\frac{7k-3}{k+1}\right)$
But P lies on the y - axis; so, its abscissa is 0.
∴ $\frac{3k-2}{k+1}=0\Rightarrow 3k-2=0\Rightarrow k=\frac{2}{3}$
So, the required ratio is $\frac{2}{3}:1$, which is 2 : 3.
$\mathrm{On}\mathrm{s}\mathrm{ubstituting}\mathrm{k}=\frac{2}{3}\mathrm{in}\left(\frac{7\mathrm{k}-3}{\mathrm{k}+1}\right),\mathrm{we}\mathrm{get}:$
$P\left(0,\frac{7\times {\displaystyle \frac{2}{3}}-3}{{\displaystyle \frac{2}{3}}+1}\right)\Rightarrow P\left(0,1\right)$
Hence, the point of intersection of AB and the y - axis is P(0, 1).