Question

A (0,0)
B (-2,3)

Determine the location of point L if it divides AB into two segments in the ratio 1:3.

a. (1/4,1/4)
b. (-1/2,3/4)
c. (-2/3,1)
d. (1,-3/2)

Answer 1

To find the point \( L \) that divides the segment \( AB \) in the ratio \( 1:3 \), we can use the section formula. The coordinates of points \( A \) and \( B \) are given as follows:

- \( A (0,0) \)
- \( B (-2,3) \)

When dividing a line segment in the ratio \( m:n \), the coordinates of the dividing point \( L \) can be calculated using the formula:

\[
L = \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right)
\]

For our case, we have \( m = 1 \) and \( n = 3 \). Substituting the coordinates of points \( A \) and \( B \):

- \( x_1 = 0 \)
- \( y_1 = 0 \)
- \( x_2 = -2 \)
- \( y_2 = 3 \)

Substituting into the section formula gives:

\[
L_x = \frac{1 \times (-2) + 3 \times 0}{1 + 3} = \frac{-2 + 0}{4} = \frac{-2}{4} = -\frac{1}{2}
\]

\[
L_y = \frac{1 \times 3 + 3 \times 0}{1 + 3} = \frac{3 + 0}{4} = \frac{3}{4}
\]

Thus, the coordinates of point \( L \) are:

\[
L = \left(-\frac{1}{2}, \frac{3}{4}\right)
\]

Therefore, the correct answer is:

**b. \((-1/2, 3/4)\)**.