Question

Write down the equation of a the line whose slope is $\frac{3}{2}$ and which passes through point P, where P divides the line segment joining A(-2, 6) and B(3, -4) in the ratio 2 : 3

Answer 1

Suppose that the coordinates of P are (x, y).
Since P divides the line segment joining the points A($-$2, 6) and B(3, $-$4) in the ratio 2 : 3, the coordinates of P are given by the formulae:

$$\begin{gathered} x=\frac{m\left(x_2\right)+n\left(x_1\right)}{m+n} \\ y=\frac{m\left(y_2\right)+n\left(y_1\right)}{m+n} \end{gathered}$$

Here, m = 2, n = 3, x₁ = $-$2, y₁ = 6, x₂ = 3 and y₂ = $-$4
By putting the values in the above formulae, we will get the coordinates of P.

$$\begin{gathered} x=\frac{2\left(3\right)+3(-2)}{2+3} \\ =\frac{6-6}{5} \\ =0 \\ y=\frac{2(-4)+3\left(6\right)}{2+3} \\ =\frac{-8+18}{5} \\ =2 \end{gathered}$$

i.e., the coordinates of point P are (0,2).
The equation of the line that has a slope and passing through P is given below:

$$\begin{gathered} y-y_1=m(x-x_1) \\ \Rightarrow y-2=\frac{3}{2}(x-0) \\ \Rightarrow 2(y-2)=3x \\ \Rightarrow 2y-4=3x \\ \Rightarrow 3x-2y+4=0 \end{gathered}$$