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: Use the section formula

If a point \(P(x,y)\) divides the line segment joining \(A(x_1,y_1)\) and \(B(x_2,y_2)\) in the ratio \(m:n\), then the coordinates of \(P\) are given by \(x=\frac{mx_2 + nx_1}{m + n}\) and \(y=\frac{my_2+ny_1}{m + n}\). Here, \(m = 3\), \(n = 2\), \(x_1=-2\), \(y_1=-4\), \(x_2 = 6\), \(y_2=1\).

Step2: Calculate the \(x\) - coordinate

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

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:

\(P(2.8,-1)\)