Question

find the coordinates of point ( p ) along the directed line segment ( ab ), from ( a(8,0) ) to ( b(3,-2) ), so that the ratio of ( ap ) to ( pb ) is 1 to 4.
( a(8,0),b(3,-2) )
the coordinates are ( p(square,square) ).

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 \(x=\frac{mx_2 + nx_1}{m + n}\) and \(y=\frac{my_2+ny_1}{m + n}\). Here \(m = 1\), \(n = 4\), \(x_1=8\), \(y_1 = 0\), \(x_2=3\), \(y_2=-2\).

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

Substitute the values into the \(x\) - formula:
\(x=\frac{1\times3+4\times8}{1 + 4}=\frac{3 + 32}{5}=\frac{35}{5}=7\)

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

Substitute the values into the \(y\) - formula:
\(y=\frac{1\times(-2)+4\times0}{1 + 4}=\frac{-2+0}{5}=-\frac{2}{5}=-0.4\)

Answer:

\(P(7,-0.4)\)