Question

1. find the point e on $overline{uv}$ with u (2, -4) and v (4, -3) such that the ratio of ue to ev is 3:4.

Explanation:

Step1: Recall the section - formula

The formula to find the coordinates of a point \(E(x,y)\) that divides the line - segment joining \(U(x_1,y_1)\) and \(V(x_2,y_2)\) in the ratio \(m:n\) is \(x=\frac{mx_2+nx_1}{m + n}\) and \(y=\frac{my_2+ny_1}{m + n}\). Here, \(x_1 = 2,y_1=-4,x_2 = 4,y_2=-3,m = 3,n = 4\).

Step2: Calculate the x - coordinate of point E

\[

$$\begin{align*} x&=\frac{3\times4 + 4\times2}{3 + 4}\\ &=\frac{12+8}{7}\\ &=\frac{20}{7} \end{align*}$$

\]

Step3: Calculate the y - coordinate of point E

\[

$$\begin{align*} y&=\frac{3\times(-3)+4\times(-4)}{3 + 4}\\ &=\frac{-9-16}{7}\\ &=\frac{-25}{7} \end{align*}$$

\]

Answer:

\(E(\frac{20}{7},-\frac{25}{7})\)