Answer :
### 1. Finding the coordinates of points which divide the line segment [tex]\( AB \)[/tex] in the given ratio
Let's use the section formula which states that the coordinates [tex]\( (x, y) \)[/tex] of a point dividing a line segment joining [tex]\( A(x_1, y_1) \)[/tex] and [tex]\( B(x_2, y_2) \)[/tex] in the ratio [tex]\( m:n \)[/tex] are given by:
[tex]\[ x = \frac{m x_2 + n x_1}{m + n} \][/tex]
[tex]\[ y = \frac{m y_2 + n y_1}{m + n} \][/tex]
Let's apply this formula to each case:
#### (a) [tex]\( A(1,1) \)[/tex] and [tex]\( B(4,4) \)[/tex]; Ratio [tex]\( 1:2 \)[/tex]
[tex]\[ x = \frac{1 \cdot 4 + 2 \cdot 1}{1 + 2} = \frac{4 + 2}{3} = \frac{6}{3} = 2 \][/tex]
[tex]\[ y = \frac{1 \cdot 4 + 2 \cdot 1}{1 + 2} = \frac{4 + 2}{3} = \frac{6}{3} = 2 \][/tex]
The coordinates are [tex]\( (2.0, 2.0) \)[/tex].
#### (b) [tex]\( A(2,6) \)[/tex] and [tex]\( B(8,3) \)[/tex]; Ratio [tex]\( 2:1 \)[/tex]
[tex]\[ x = \frac{2 \cdot 8 + 1 \cdot 2}{2 + 1} = \frac{16 + 2}{3} = \frac{18}{3} = 6 \][/tex]
[tex]\[ y = \frac{2 \cdot 3 + 1 \cdot 6}{2 + 1} = \frac{6 + 6}{3} = \frac{12}{3} = 4 \][/tex]
The coordinates are [tex]\( (6.0, 4.0) \)[/tex].
#### (c) [tex]\( A(4,5) \)[/tex] and [tex]\( B(12,1) \)[/tex]; Ratio [tex]\( 3:1 \)[/tex]
[tex]\[ x = \frac{3 \cdot 12 + 1 \cdot 4}{3 + 1} = \frac{36 + 4}{4} = \frac{40}{4} = 10 \][/tex]
[tex]\[ y = \frac{3 \cdot 1 + 1 \cdot 5}{3 + 1} = \frac{3 + 5}{4} = \frac{8}{4} = 2 \][/tex]
The coordinates are [tex]\( (10.0, 2.0) \)[/tex].
#### (d) [tex]\( A(2,6) \)[/tex] and [tex]\( B(12,1) \)[/tex]; Ratio [tex]\( 3:2 \)[/tex]
[tex]\[ x = \frac{3 \cdot 12 + 2 \cdot 2}{3 + 2} = \frac{36 + 4}{5} = \frac{40}{5} = 8 \][/tex]
[tex]\[ y = \frac{3 \cdot 1 + 2 \cdot 6}{3 + 2} = \frac{3 + 12}{5} = \frac{15}{5} = 3 \][/tex]
The coordinates are [tex]\( (8.0, 3.0) \)[/tex].
#### (e) [tex]\( A(-7,5) \)[/tex] and [tex]\( B(3,0) \)[/tex]; Ratio [tex]\( 3:2 \)[/tex]
[tex]\[ x = \frac{3 \cdot 3 + 2 \cdot (-7)}{3 + 2} = \frac{9 + (-14)}{5} = \frac{-5}{5} = -1 \][/tex]
[tex]\[ y = \frac{3 \cdot 0 + 2 \cdot 5}{3 + 2} = \frac{0 + 10}{5} = \frac{10}{5} = 2 \][/tex]
The coordinates are [tex]\( (-1.0, 2.0) \)[/tex].
#### (f) [tex]\( A(-7,5) \)[/tex] and [tex]\( B(5,-1) \)[/tex]; Ratio [tex]\( 5:1 \)[/tex]
[tex]\[ x = \frac{5 \cdot 5 + 1 \cdot (-7)}{5 + 1} = \frac{25 + (-7)}{6} = \frac{18}{6} = 3 \][/tex]
[tex]\[ y = \frac{5 \cdot (-1) + 1 \cdot 5}{5 + 1} = \frac{-5 + 5}{6} = \frac{0}{6} = 0 \][/tex]
The coordinates are [tex]\( (3.0, 0.0) \)[/tex].
### 2. Finding the ratio in which the point [tex]\( P \)[/tex] divides [tex]\( AB \)[/tex] internally
To find the ratio [tex]\( m:n \)[/tex] in which point [tex]\( P(x_p, y_p) \)[/tex] divides the line segment [tex]\( A(x_1, y_1) \)[/tex] and [tex]\( B(x_2, y_2) \)[/tex], we use the formulas:
[tex]\[ m = \frac{x_p - x_1}{x_2 - x_1} \][/tex]
[tex]\[ n = \frac{y_p - y_1}{y_2 - y_1} \][/tex]
#### (a) [tex]\( A(1,1) \)[/tex], [tex]\( B(4,4) \)[/tex], and [tex]\( P(2,2) \)[/tex]
[tex]\[ m = \frac{2 - 1}{4 - 1} = \frac{1}{3} \][/tex]
[tex]\[ n = \frac{2 - 1}{4 - 1} = \frac{1}{3} \][/tex]
The ratio is [tex]\( \left( \frac{1}{3}, \frac{1}{3} \right) \)[/tex] or [tex]\( 0.3333\ldots \)[/tex].
#### (b) [tex]\( A(2,6) \)[/tex], [tex]\( B(8,3) \)[/tex], and [tex]\( P(6,4) \)[/tex]
[tex]\[ m = \frac{6 - 2}{8 - 2} = \frac{4}{6} = \frac{2}{3} \][/tex]
[tex]\[ n = \frac{4 - 6}{3 - 6} = \frac{-2}{-3} = \frac{2}{3} \][/tex]
The ratio is [tex]\( \left( \frac{2}{3}, \frac{2}{3} \right) \)[/tex] or [tex]\( 0.6666\ldots \)[/tex].
#### (c) [tex]\( A(-3,3) \)[/tex], [tex]\( B(5,-1) \)[/tex], and [tex]\( P(3,0) \)[/tex]
[tex]\[ m = \frac{3 - (-3)}{5 - (-3)} = \frac{6}{8} = \frac{3}{4} \][/tex]
[tex]\[ n = \frac{0 - 3}{-1 - 3} = \frac{-3}{-4} = \frac{3}{4} \][/tex]
The ratio is [tex]\( \left( \frac{3}{4}, \frac{3}{4} \right) \)[/tex] or [tex]\( 0.75 \)[/tex].
#### (d) [tex]\( A(-3,3) \)[/tex], [tex]\( B(5,-1) \)[/tex], and [tex]\( P(-1,2) \)[/tex]
[tex]\[ m = \frac{-1 - (-3)}{5 - (-3)} = \frac{2}{8} = \frac{1}{4} \][/tex]
[tex]\[ n = \frac{2 - 3}{-1 - 3} = \frac{-1}{-4} = \frac{1}{4} \][/tex]
The ratio is [tex]\( \left( \frac{1}{4}, \frac{1}{4} \right) \)[/tex] or [tex]\( 0.25 \)[/tex].
#### (e) [tex]\( A(2,6) \)[/tex], [tex]\( B(12,1) \)[/tex], and [tex]\( P(10,2) \)[/tex]
[tex]\[ m = \frac{10 - 2}{12 - 2} = \frac{8}{10} = \frac{4}{5} \][/tex]
[tex]\[ n = \frac{2 - 6}{1 - 6} = \frac{-4}{-5} = \frac{4}{5} \][/tex]
The ratio is [tex]\( \left( \frac{4}{5}, \frac{4}{5} \right) \)[/tex] or [tex]\( 0.8 \)[/tex].
#### (f) [tex]\( A(2,6) \)[/tex], [tex]\( B(14,0) \)[/tex], and [tex]\( P(12,1) \)[/tex]
[tex]\[ m = \frac{12 - 2}{14 - 2} = \frac{10}{12} = \frac{5}{6} \][/tex]
[tex]\[ n = \frac{1 - 6}{0 - 6} = \frac{-5}{-6} = \frac{5}{6} \][/tex]
The ratio is [tex]\( \left( \frac{5}{6}, \frac{5}{6} \right) \)[/tex] or [tex]\( 0.8333\ldots \)[/tex].
Let's use the section formula which states that the coordinates [tex]\( (x, y) \)[/tex] of a point dividing a line segment joining [tex]\( A(x_1, y_1) \)[/tex] and [tex]\( B(x_2, y_2) \)[/tex] in the ratio [tex]\( m:n \)[/tex] are given by:
[tex]\[ x = \frac{m x_2 + n x_1}{m + n} \][/tex]
[tex]\[ y = \frac{m y_2 + n y_1}{m + n} \][/tex]
Let's apply this formula to each case:
#### (a) [tex]\( A(1,1) \)[/tex] and [tex]\( B(4,4) \)[/tex]; Ratio [tex]\( 1:2 \)[/tex]
[tex]\[ x = \frac{1 \cdot 4 + 2 \cdot 1}{1 + 2} = \frac{4 + 2}{3} = \frac{6}{3} = 2 \][/tex]
[tex]\[ y = \frac{1 \cdot 4 + 2 \cdot 1}{1 + 2} = \frac{4 + 2}{3} = \frac{6}{3} = 2 \][/tex]
The coordinates are [tex]\( (2.0, 2.0) \)[/tex].
#### (b) [tex]\( A(2,6) \)[/tex] and [tex]\( B(8,3) \)[/tex]; Ratio [tex]\( 2:1 \)[/tex]
[tex]\[ x = \frac{2 \cdot 8 + 1 \cdot 2}{2 + 1} = \frac{16 + 2}{3} = \frac{18}{3} = 6 \][/tex]
[tex]\[ y = \frac{2 \cdot 3 + 1 \cdot 6}{2 + 1} = \frac{6 + 6}{3} = \frac{12}{3} = 4 \][/tex]
The coordinates are [tex]\( (6.0, 4.0) \)[/tex].
#### (c) [tex]\( A(4,5) \)[/tex] and [tex]\( B(12,1) \)[/tex]; Ratio [tex]\( 3:1 \)[/tex]
[tex]\[ x = \frac{3 \cdot 12 + 1 \cdot 4}{3 + 1} = \frac{36 + 4}{4} = \frac{40}{4} = 10 \][/tex]
[tex]\[ y = \frac{3 \cdot 1 + 1 \cdot 5}{3 + 1} = \frac{3 + 5}{4} = \frac{8}{4} = 2 \][/tex]
The coordinates are [tex]\( (10.0, 2.0) \)[/tex].
#### (d) [tex]\( A(2,6) \)[/tex] and [tex]\( B(12,1) \)[/tex]; Ratio [tex]\( 3:2 \)[/tex]
[tex]\[ x = \frac{3 \cdot 12 + 2 \cdot 2}{3 + 2} = \frac{36 + 4}{5} = \frac{40}{5} = 8 \][/tex]
[tex]\[ y = \frac{3 \cdot 1 + 2 \cdot 6}{3 + 2} = \frac{3 + 12}{5} = \frac{15}{5} = 3 \][/tex]
The coordinates are [tex]\( (8.0, 3.0) \)[/tex].
#### (e) [tex]\( A(-7,5) \)[/tex] and [tex]\( B(3,0) \)[/tex]; Ratio [tex]\( 3:2 \)[/tex]
[tex]\[ x = \frac{3 \cdot 3 + 2 \cdot (-7)}{3 + 2} = \frac{9 + (-14)}{5} = \frac{-5}{5} = -1 \][/tex]
[tex]\[ y = \frac{3 \cdot 0 + 2 \cdot 5}{3 + 2} = \frac{0 + 10}{5} = \frac{10}{5} = 2 \][/tex]
The coordinates are [tex]\( (-1.0, 2.0) \)[/tex].
#### (f) [tex]\( A(-7,5) \)[/tex] and [tex]\( B(5,-1) \)[/tex]; Ratio [tex]\( 5:1 \)[/tex]
[tex]\[ x = \frac{5 \cdot 5 + 1 \cdot (-7)}{5 + 1} = \frac{25 + (-7)}{6} = \frac{18}{6} = 3 \][/tex]
[tex]\[ y = \frac{5 \cdot (-1) + 1 \cdot 5}{5 + 1} = \frac{-5 + 5}{6} = \frac{0}{6} = 0 \][/tex]
The coordinates are [tex]\( (3.0, 0.0) \)[/tex].
### 2. Finding the ratio in which the point [tex]\( P \)[/tex] divides [tex]\( AB \)[/tex] internally
To find the ratio [tex]\( m:n \)[/tex] in which point [tex]\( P(x_p, y_p) \)[/tex] divides the line segment [tex]\( A(x_1, y_1) \)[/tex] and [tex]\( B(x_2, y_2) \)[/tex], we use the formulas:
[tex]\[ m = \frac{x_p - x_1}{x_2 - x_1} \][/tex]
[tex]\[ n = \frac{y_p - y_1}{y_2 - y_1} \][/tex]
#### (a) [tex]\( A(1,1) \)[/tex], [tex]\( B(4,4) \)[/tex], and [tex]\( P(2,2) \)[/tex]
[tex]\[ m = \frac{2 - 1}{4 - 1} = \frac{1}{3} \][/tex]
[tex]\[ n = \frac{2 - 1}{4 - 1} = \frac{1}{3} \][/tex]
The ratio is [tex]\( \left( \frac{1}{3}, \frac{1}{3} \right) \)[/tex] or [tex]\( 0.3333\ldots \)[/tex].
#### (b) [tex]\( A(2,6) \)[/tex], [tex]\( B(8,3) \)[/tex], and [tex]\( P(6,4) \)[/tex]
[tex]\[ m = \frac{6 - 2}{8 - 2} = \frac{4}{6} = \frac{2}{3} \][/tex]
[tex]\[ n = \frac{4 - 6}{3 - 6} = \frac{-2}{-3} = \frac{2}{3} \][/tex]
The ratio is [tex]\( \left( \frac{2}{3}, \frac{2}{3} \right) \)[/tex] or [tex]\( 0.6666\ldots \)[/tex].
#### (c) [tex]\( A(-3,3) \)[/tex], [tex]\( B(5,-1) \)[/tex], and [tex]\( P(3,0) \)[/tex]
[tex]\[ m = \frac{3 - (-3)}{5 - (-3)} = \frac{6}{8} = \frac{3}{4} \][/tex]
[tex]\[ n = \frac{0 - 3}{-1 - 3} = \frac{-3}{-4} = \frac{3}{4} \][/tex]
The ratio is [tex]\( \left( \frac{3}{4}, \frac{3}{4} \right) \)[/tex] or [tex]\( 0.75 \)[/tex].
#### (d) [tex]\( A(-3,3) \)[/tex], [tex]\( B(5,-1) \)[/tex], and [tex]\( P(-1,2) \)[/tex]
[tex]\[ m = \frac{-1 - (-3)}{5 - (-3)} = \frac{2}{8} = \frac{1}{4} \][/tex]
[tex]\[ n = \frac{2 - 3}{-1 - 3} = \frac{-1}{-4} = \frac{1}{4} \][/tex]
The ratio is [tex]\( \left( \frac{1}{4}, \frac{1}{4} \right) \)[/tex] or [tex]\( 0.25 \)[/tex].
#### (e) [tex]\( A(2,6) \)[/tex], [tex]\( B(12,1) \)[/tex], and [tex]\( P(10,2) \)[/tex]
[tex]\[ m = \frac{10 - 2}{12 - 2} = \frac{8}{10} = \frac{4}{5} \][/tex]
[tex]\[ n = \frac{2 - 6}{1 - 6} = \frac{-4}{-5} = \frac{4}{5} \][/tex]
The ratio is [tex]\( \left( \frac{4}{5}, \frac{4}{5} \right) \)[/tex] or [tex]\( 0.8 \)[/tex].
#### (f) [tex]\( A(2,6) \)[/tex], [tex]\( B(14,0) \)[/tex], and [tex]\( P(12,1) \)[/tex]
[tex]\[ m = \frac{12 - 2}{14 - 2} = \frac{10}{12} = \frac{5}{6} \][/tex]
[tex]\[ n = \frac{1 - 6}{0 - 6} = \frac{-5}{-6} = \frac{5}{6} \][/tex]
The ratio is [tex]\( \left( \frac{5}{6}, \frac{5}{6} \right) \)[/tex] or [tex]\( 0.8333\ldots \)[/tex].
