To find the coordinates of point P that partitions the segment AB in the ratio of 2:5, we use the section formula. The section formula for the coordinates of point P dividing the line segment joining points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] in the ratio [tex]\(m:n\)[/tex] is given by:
[tex]\[
\left( \frac{mx_2 + nx_1}{m + n}, \frac{my_2 + ny_1}{m + n} \right)
\][/tex]
Given:
- Point A has coordinates [tex]\( (x_1, y_1) = (3, 2) \)[/tex]
- Point B has coordinates [tex]\( (x_2, y_2) = (3, 16) \)[/tex]
- The ratio [tex]\( m:n \)[/tex] is [tex]\( 2:5 \)[/tex]
First, calculate the x-coordinate of point P:
[tex]\[
Px = \frac{m \cdot x_2 + n \cdot x_1}{m + n}
\][/tex]
Substitute the given values:
[tex]\[
Px = \frac{2 \cdot 3 + 5 \cdot 3}{2 + 5} = \frac{6 + 15}{7} = \frac{21}{7} = 3
\][/tex]
Now, calculate the y-coordinate of point P:
[tex]\[
Py = \frac{m \cdot y_2 + n \cdot y_1}{m + n}
\][/tex]
Substitute the given values:
[tex]\[
Py = \frac{2 \cdot 16 + 5 \cdot 2}{2 + 5} = \frac{32 + 10}{7} = \frac{42}{7} = 6
\][/tex]
Therefore, the coordinates of point P are [tex]\( (3, 6) \)[/tex].
Among the given choices:
1. (3,4)
2. (3,6)
3. (3,12)
4. (3,14)
The correct answer is: (3,6).
