Question

find the coordinates of point p along the directed line segment ab, from a(-2, -4) to b(6, 1), so that the ratio of ap to pb is 3 to 2. the coordinates are p( , ).

Explanation:

Step1: Recall the section - formula

The formula for finding 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 given by \(x=\frac{mx_2+nx_1}{m + n}\) and \(y=\frac{my_2+ny_1}{m + n}\). Here, \(x_1=-2,y_1=-4,x_2 = 6,y_2 = 1,m = 3,n = 2\).

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

Substitute the values into the \(x\) - coordinate formula:
\[x=\frac{3\times6+2\times(-2)}{3 + 2}=\frac{18-4}{5}=\frac{14}{5}=2.8\]

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

Substitute the values into the \(y\) - coordinate formula:
\[y=\frac{3\times1+2\times(-4)}{3 + 2}=\frac{3-8}{5}=\frac{-5}{5}=-1\]

Answer:

\((2.8,-1)\)