Find the ratio in which the line segment joining (−2, −3)

Question:

Find the ratio in which the line segment joining (−2, −3) and (5, 6) is divided by (i) x-axis (ii) y-axis. Also, find the coordinates of the point of division in each case.

Solution:

The ratio in which the $x$-axis divides two points $\left(x_{1}, y_{1}\right)$ and $\left(x_{2}, y_{2}\right)$ is $\lambda: 1$

The ratio in which the $y$-axis divides two points $\left(x_{1}, y_{1}\right)$ and $\left(x_{2}, y_{2}\right)$ is $\mu: 1$

The co-ordinates of the point dividing two points $\left(x_{1}, y_{1}\right)$ and $\left(x_{2}, y_{2}\right)$ in the ratio $m: n$ is given as,

$(x, y)=\left(\left(\frac{\lambda x_{2}+x_{1}}{\lambda+1}\right),\left(\frac{\lambda y_{2}+y_{1}}{\lambda+1}\right)\right)$ Where $\lambda=\frac{m}{n}$

Here the two given points are $A(-2,-3)$ and $B(5,6)$.

i. The ratio in which the $x$-axis divides these points is

$\frac{6 \lambda-3}{3}=0$

$\lambda=\frac{1}{2}$

Let point P(x, y) divide the line joining ‘AB’ in the ratio 

Substituting these values in the earlier mentioned formula we have,

$\left.(x, y)=\left(\frac{\frac{1}{2}(5)+(-2)}{\frac{1}{2}+1}\right),\left(\frac{\frac{1}{2}(6)+(-3)}{\frac{1}{2}+1}\right)\right)$

$(x, y)=\left(\left(\frac{\frac{5+2(-2)}{2}}{\frac{1+2}{2}}\right),\left(\frac{\frac{6+2(-3)}{2}}{\frac{1+2}{2}}\right)\right)$

$(x, y)=\left(\left(\frac{1}{3}\right),\left(\frac{0}{3}\right)\right)$

$(x, y)=\left(\frac{1}{3}, 0\right)$

Thus the ratio in which the $x$-axis divides the two given points and the co-ordinates of the point is $\left(\begin{array}{l}1: 2 \\ \left(\frac{1}{3}, 0\right)\end{array}\right)$.

ii. The ratio in which the $y$-axis divides these points is

$\frac{5 \mu-2}{3}=0$

$\Rightarrow \mu=\frac{2}{5}$

Let point P(x, y) divide the line joining ‘AB’ in the ratio 

Substituting these values in the earlier mentioned formula we have,

$(x, y)=\left(\left(\frac{\frac{2}{5}(5)+(-2)}{\frac{2}{5}+1}\right) \cdot\left(\frac{\frac{2}{5}(6)+(-3)}{\frac{2}{5}+1}\right)\right)$

$(x, y)=\left(\left(\frac{\frac{10+5(-2)}{5}}{\frac{2+5}{5}}\right),\left(\frac{\frac{12+5(-3)}{5}}{\frac{2+5}{5}}\right)\right)$

$(x, y)=\left(\left(\frac{0}{7}\right),\left(-\frac{3}{7}\right)\right)$

$(x, y)=\left(0,-\frac{3}{7}\right)$

Thus the ratio in which the $x$-axis divides the two given points and the co-ordinates of the point is $2: 5$ and $\left(0,-\frac{3}{7}\right)$.

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