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
Let y-axis divides PQ in the ratio.$\lambda:1$
Let R(x, y) be the coordinates of the point of division.
Then, the coordinates of the point of division are,$\text{R}\bigg(\frac{3\times\lambda+(-2)\times1}{\lambda+1},\frac{7\times\lambda+(-3)\times1}{\lambda+1}\bigg)=\bigg(\frac{3\lambda-2}{\lambda+1},\frac{7\lambda-3}{\lambda+1}\bigg)$
Since R lies on y-axis and x-coordinates of every point on y-axis is zero.$\therefore\ \frac{3\lambda-2}{\lambda+1}=0$
$\Rightarrow\ 3\lambda-2=0$
$\Rightarrow\ 3\lambda=2$
$\Rightarrow\ \lambda=\frac{2}{3}$
Hence, the required ratio is $\frac{2}{3}:1$ i.e., 2 : 3 putting $\lambda=\frac{2}{3}$ in the coordinates of R, we get$\Rightarrow\ \frac{7\lambda-3}{\lambda+1}=\frac{7\times\frac{2}{3}-3}{\frac{2}{3}+1}$
$\Rightarrow\ \frac{\frac{14-9}{3}}{\frac{2+3}{3}}=\frac{\frac{5}{3}}{\frac{5}{3}}=1$
Hence, the coordinates of R(0, 1).
Let R(x, y) be the coordinates of the point of division.
Then, the coordinates of the point of division are,$\text{R}\bigg(\frac{3\times\lambda+(-2)\times1}{\lambda+1},\frac{7\times\lambda+(-3)\times1}{\lambda+1}\bigg)=\bigg(\frac{3\lambda-2}{\lambda+1},\frac{7\lambda-3}{\lambda+1}\bigg)$
Since R lies on y-axis and x-coordinates of every point on y-axis is zero.$\therefore\ \frac{3\lambda-2}{\lambda+1}=0$
$\Rightarrow\ 3\lambda-2=0$
$\Rightarrow\ 3\lambda=2$
$\Rightarrow\ \lambda=\frac{2}{3}$
Hence, the required ratio is $\frac{2}{3}:1$ i.e., 2 : 3 putting $\lambda=\frac{2}{3}$ in the coordinates of R, we get$\Rightarrow\ \frac{7\lambda-3}{\lambda+1}=\frac{7\times\frac{2}{3}-3}{\frac{2}{3}+1}$
$\Rightarrow\ \frac{\frac{14-9}{3}}{\frac{2+3}{3}}=\frac{\frac{5}{3}}{\frac{5}{3}}=1$
Hence, the coordinates of R(0, 1).
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