Question

Point C (4, 2) divides the line segment joining points A (2, -1) and B (x, y) such that AC: CB = 3.1 what are the coordinates of point B?

Answer 1

(14/3 ,3)

Explanation:

Answer 2

The coordinates of B are
`((14)/(3) ,3)`.

Solution:

Given A(2, –1), B(x, y) and C(4, 2)

AB is a line segment C is a point on AB.

AC : CB = 3 : 1

To find the coordinates of B:

Section formula:

The point
`P(x,y)` divides the line segment
`A(x_1,y_1)` and
`B(x_2,y_2)` in the ratio

m : n are
`\left(\frac{\mathbf{m} \mathbf{x}_(2)+\mathbf{n} \mathbf{x}_{\mathbf{1}}}{\mathbf{m}+\mathbf{n}}, \frac{\mathbf{m} \mathbf{y}_(2)+\mathbf{n} \mathbf{y}_{\mathbf{1}}}{\mathbf{m}+\mathbf{n}}\right)`

Here,
`x_1=2, \ x_2=x, \ x_3=4, \ y_1=-1, \ y_2=y, \ y_3=2` and m = 3, n = 1

Using section formula,

`$C(x_3,y_3)=\left((mx_(2)+n x_(1))/(m+n), (m y_(2)+n y_(1))/(m+n)\right)`

Substitute the given values in the section formula.

`$C(4,2)=\left((3 * x+1 * 2)/(3+1), (3 * y+1 * (-1))/(3+1)\right)`

`$C(4,2)=\left((3x+2)/(4), (3y-1)/(4)\right)`

Equate the x-coordinates and y-coordinates.

`$(3x+2)/(4)=4, \ \ \ (3y-1)/(4)=2`

`$\ 3x+2=16, \ \ \ \ 3y-1=8`

`$ 3x=14, \ \ \ \ 3y=9`

`$ x=(14)/(3), \ \ \ \ y=3`

Hence the coordinates of B are
`((14)/(3) ,3)`.