Let's solve the problem step-by-step.
### Problem Statement
We are given two points, [tex]\( A(0, -3) \)[/tex] and [tex]\( B(7, 1) \)[/tex], and we need to find the point [tex]\( P \)[/tex] that divides the line segment [tex]\( AB \)[/tex] internally in the specified ratios.
#### Part (a): Ratio 2:1
We need to find the coordinates of point [tex]\( P \)[/tex] that divides the line segment joining [tex]\( A(0, -3) \)[/tex] and [tex]\( B(7, 1) \)[/tex] in the ratio 2:1.
To find this, we use the section formula for internal division. For points [tex]\( A(x_1, y_1) \)[/tex] and [tex]\( B(x_2, y_2) \)[/tex] dividing the segment in the ratio [tex]\( m:n \)[/tex], the coordinates of point [tex]\( P \)[/tex] are given by:
[tex]\[
P \left( \frac{m x_2 + n x_1}{m + n}, \frac{m y_2 + n y_1}{m + n} \right)
\][/tex]
Here, [tex]\( m = 2 \)[/tex] and [tex]\( n = 1 \)[/tex]. Substituting [tex]\( A \)[/tex] as [tex]\( (0, -3) \)[/tex] and [tex]\( B \)[/tex] as [tex]\( (7, 1) \)[/tex]:
[tex]\[
P_x = \frac{2 \cdot 7 + 1 \cdot 0}{2 + 1} = \frac{14 + 0}{3} = \frac{14}{3} = 4.666666666666667
\][/tex]
[tex]\[
P_y = \frac{2 \cdot 1 + 1 \cdot -3}{2 + 1} = \frac{2 - 3}{3} = \frac{-1}{3} = -0.3333333333333333
\][/tex]
So, the coordinates of point [tex]\( P \)[/tex] in Part (a) are [tex]\( \left( 4.666666666666667, -0.3333333333333333 \right) \)[/tex].
#### Part (b): Ratio 1:1
We need to find the coordinates of point [tex]\( P \)[/tex] that divides the line segment joining [tex]\( A(0, -3) \)[/tex] and [tex]\( B(7, 1) \)[/tex] in the ratio 1:1.
For the ratio 1:1, point [tex]\( P \)[/tex] is essentially the midpoint of [tex]\( A \)[/tex] and [tex]\( B \)[/tex]. The midpoint [tex]\( P \)[/tex] of [tex]\( A(x_1, y_1) \)[/tex] and [tex]\( B(x_2, y_2) \)[/tex] is given by:
[tex]\[
P \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
\][/tex]
Substituting [tex]\( A \)[/tex] as [tex]\( (0, -3) \)[/tex] and [tex]\( B \)[/tex] as [tex]\( (7, 1) \)[/tex]:
[tex]\[
P_x = \frac{0 + 7}{2} = \frac{7}{2} = 3.5
\][/tex]
[tex]\[
P_y = \frac{-3 + 1}{2} = \frac{-2}{2} = -1
\][/tex]
So, the coordinates of point [tex]\( P \)[/tex] in Part (b) are [tex]\( (3.5, -1) \)[/tex].
### Summary
- Part (a): The coordinates of point [tex]\( P \)[/tex] dividing [tex]\( AB \)[/tex] in the ratio 2:1 are [tex]\( \left( 4.666666666666667, -0.3333333333333333 \right) \)[/tex].
- Part (b): The coordinates of point [tex]\( P \)[/tex] dividing [tex]\( AB \)[/tex] in the ratio 1:1 (midpoint) are [tex]\( (3.5, -1) \)[/tex].
