Question

1. given ab with a(8, - 4) and b(-6, -11), if p partitions ab such that the ratio of ap to ab is 2:7, find the coordinates of p.

Explanation:

Step1: Recall the section - formula

The coordinates of a point \(P(x,y)\) that divides the line - segment joining \(A(x_1,y_1)\) and \(B(x_2,y_2)\) in the ratio \(m:n\) are given by \(x=\frac{mx_2+nx_1}{m + n}\) and \(y=\frac{my_2+ny_1}{m + n}\). Here, \(A(8,-4)\), \(B(-6,-11)\), and the ratio \(m:n = 2:7\).

Step2: Calculate the \(x\) - coordinate of \(P\)

\[

$$\begin{align*} x&=\frac{2\times(-6)+7\times8}{2 + 7}\\ &=\frac{-12 + 56}{9}\\ &=\frac{44}{9} \end{align*}$$

\]

Step3: Calculate the \(y\) - coordinate of \(P\)

\[

$$\begin{align*} y&=\frac{2\times(-11)+7\times(-4)}{2 + 7}\\ &=\frac{-22-28}{9}\\ &=\frac{-50}{9} \end{align*}$$

\]

Answer:

\(P(\frac{44}{9},-\frac{50}{9})\)