Question

what are the coordinates of the point on the directed line segment from (-5,1) to (2,-6) that partitions the segment into a ratio of 4 to 3?

Explanation:

Step1: Recall the section - formula

The formula to find 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 \(m:n\) is \(x=\frac{mx_2+nx_1}{m + n}\) and \(y=\frac{my_2+ny_1}{m + n}\). Here, \(x_1=-5,y_1 = 1,x_2=2,y_2=-6,m = 4,n = 3\).

Step2: Calculate the \(x\) - coordinate

Substitute the values into the \(x\) - coordinate formula:
\[

$$\begin{align*} x&=\frac{4\times2+3\times(-5)}{4 + 3}\\ &=\frac{8-15}{7}\\ &=\frac{-7}{7}\\ &=-1 \end{align*}$$

\]

Step3: Calculate the \(y\) - coordinate

Substitute the values into the \(y\) - coordinate formula:
\[

$$\begin{align*} y&=\frac{4\times(-6)+3\times1}{4 + 3}\\ &=\frac{-24 + 3}{7}\\ &=\frac{-21}{7}\\ &=-3 \end{align*}$$

\]

Answer:

\((-1,-3)\)