Question

What are the coordinates of point E such that AP is 3/4 of the distance from A to E

Answer 1

Given:

The coordinates of point P(-4,0).

The coordinates of point A(2,-6).

Step-by-step explanation:

The distance of AP is 3/4 times of AE. So,

`\begin{gathered} AP=(3)/(4)AE \\ (AE)/(AP)=(4)/(3) \\ (AP+PE)/(AP)=(4)/(3) \\ (PE)/(AP)=(4)/(3)-1 \\ (PE)/(AP)=(1)/(3) \end{gathered}`

The formula for the coordinates of the point lying between endpoint (x_1,y_1) and (x_2,y_2) such that divide the line in m:n is,

`(x,y)=((mx_2+nx_1)/(m+n),(my_2+ny_1)/(m+n))`

So coordinates for point P lying on segment AE is,

`\begin{gathered} (-4,0)=((3\cdot x+1\cdot2)/(1+3),(3\cdot y+1\cdot(-6))/(1+3)) \\ =((3x+2)/(4),(3y-6)/(4)) \end{gathered}`

Solve the equations for x and y.

`\begin{gathered} (3x+2)/(4)=-4 \\ 3x+2=-16 \\ x=-(18)/(3) \\ =-6 \end{gathered}`

For y,

`\begin{gathered} (3y-6)/(4)=0 \\ 3y=6 \\ y=(6)/(3) \\ =2 \end{gathered}`

So coordinates of point E is (-6,2).