Answer :
Let's solve this step-by-step.
### Part 1: Finding the coordinates of the point which divides the line segment AB in the given ratio
Given two points [tex]\( A(x_1, y_1) \)[/tex] and [tex]\( B(x_2, y_2) \)[/tex], and a ratio [tex]\( m:n \)[/tex], the coordinates of the point [tex]\( P(x, y) \)[/tex] that divides AB in the ratio [tex]\( m:n \)[/tex] can be calculated using the section formula:
[tex]\[ x = \frac{mx_2 + nx_1}{m+n} \][/tex]
[tex]\[ y = \frac{my_2 + ny_1}{m+n} \][/tex]
#### (a) [tex]\( A = (1,1) \)[/tex] and [tex]\( B = (4,4) \)[/tex]; Ratio [tex]\( = 1 : 2 \)[/tex]
[tex]\[ m = 1, n = 2 \][/tex]
[tex]\[ x = \frac{1 \cdot 4 + 2 \cdot 1}{1+2} = \frac{4 + 2}{3} = 2.0 \][/tex]
[tex]\[ y = \frac{1 \cdot 4 + 2 \cdot 1}{1+2} = \frac{4 + 2}{3} = 2.0 \][/tex]
So 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]\[ m = 2, n = 1 \][/tex]
[tex]\[ x = \frac{2 \cdot 8 + 1 \cdot 2}{2+1} = \frac{16 + 2}{3} = 6.0 \][/tex]
[tex]\[ y = \frac{2 \cdot 3 + 1 \cdot 6}{2+1} = \frac{6 + 6}{3} = 4.0 \][/tex]
So 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]\[ m = 3, n = 1 \][/tex]
[tex]\[ x = \frac{3 \cdot 12 + 1 \cdot 4}{3+1} = \frac{36 + 4}{4} = 10.0 \][/tex]
[tex]\[ y = \frac{3 \cdot 1 + 1 \cdot 5}{3+1} = \frac{3 + 5}{4} = 2.0 \][/tex]
So 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]\[ m = 3, n = 2 \][/tex]
[tex]\[ x = \frac{3 \cdot 12 + 2 \cdot 2}{3+2} = \frac{36 + 4}{5} = 8.0 \][/tex]
[tex]\[ y = \frac{3 \cdot 1 + 2 \cdot 6}{3+2} = \frac{3 + 12}{5} = 3.0 \][/tex]
So 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]\[ m = 3, n = 2 \][/tex]
[tex]\[ x = \frac{3 \cdot 3 + 2 \cdot (-7)}{3+2} = \frac{9 - 14}{5} = -1.0 \][/tex]
[tex]\[ y = \frac{3 \cdot 0 + 2 \cdot 5}{3+2} = \frac{0 + 10}{5} = 2.0 \][/tex]
So 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]\[ m = 5, n = 1 \][/tex]
[tex]\[ x = \frac{5 \cdot 5 + 1 \cdot (-7)}{5+1} = \frac{25 - 7}{6} = 3.0 \][/tex]
[tex]\[ y = \frac{5 \cdot (-1) + 1 \cdot 5}{5+1} = \frac{-5 + 5}{6} = 0.0 \][/tex]
So the coordinates are [tex]\( (3.0, 0.0) \)[/tex].
### Part 2: Finding the ratio in which the point P divides AB internally
Given points [tex]\( A(x_1, y_1) \)[/tex], [tex]\( B(x_2, y_2) \)[/tex], and [tex]\( P(x, y) \)[/tex], the ratio [tex]\( m:n \)[/tex] in which P divides AB can be calculated using:
[tex]\[ m : n = \frac{x_2 - x}{x - x_1} \][/tex]
#### (a) [tex]\( A = (1,1) \)[/tex], [tex]\( B = (4,4) \)[/tex], and [tex]\( P = (2,2) \)[/tex]
[tex]\[ \frac{x_2 - x}{x - x_1} = \frac{4 - 2}{2 - 1} = \frac{2}{1} = 2 \][/tex]
So the ratio is [tex]\( 2 : 1 \)[/tex].
#### (b) [tex]\( A = (2,6) \)[/tex], [tex]\( B = (8,3) \)[/tex], and [tex]\( P = (6,4) \)[/tex]
[tex]\[ \frac{x_2 - x}{x - x_1} = \frac{8 - 6}{6 - 2} = \frac{2}{4} = 0.5 \][/tex]
So the ratio is [tex]\( 0.5 : 1 \)[/tex].
#### (c) [tex]\( A = (-3,3) \)[/tex], [tex]\( B = (5,-1) \)[/tex], and [tex]\( P = (3,0) \)[/tex]
[tex]\[ \frac{x_2 - x}{x - x_1} = \frac{5 - 3}{3 - (-3)} = \frac{2}{6} \approx 0.33 \][/tex]
So the ratio is [tex]\( 0.33 : 1 \)[/tex].
#### (d) [tex]\( A = (-3,3) \)[/tex], [tex]\( B = (5,-1) \)[/tex], and [tex]\( P = (-1,2) \)[/tex]
[tex]\[ \frac{x_2 - x}{x - x_1} = \frac{5 - (-1)}{-1 - (-3)} = \frac{6}{2} = 3 \][/tex]
So the ratio is [tex]\( 3 : 1 \)[/tex].
#### (e) [tex]\( A = (2,6) \)[/tex], [tex]\( B = (12,1) \)[/tex], and [tex]\( P = (10,2) \)[/tex]
[tex]\[ \frac{x_2 - x}{x - x_1} = \frac{12 - 10}{10 - 2} = \frac{2}{8} = 0.25 \][/tex]
So the ratio is [tex]\( 0.25 : 1 \)[/tex].
#### (f) [tex]\( A = (2,6) \)[/tex], [tex]\( B = (14,0) \)[/tex], and [tex]\( P = (12,1) \)[/tex]
[tex]\[ \frac{x_2 - x}{x - x_1} = \frac{14 - 12}{12 - 2} = \frac{2}{10} = 0.2 \][/tex]
So the ratio is [tex]\( 0.2 : 1 \)[/tex].
### Part 1: Finding the coordinates of the point which divides the line segment AB in the given ratio
Given two points [tex]\( A(x_1, y_1) \)[/tex] and [tex]\( B(x_2, y_2) \)[/tex], and a ratio [tex]\( m:n \)[/tex], the coordinates of the point [tex]\( P(x, y) \)[/tex] that divides AB in the ratio [tex]\( m:n \)[/tex] can be calculated using the section formula:
[tex]\[ x = \frac{mx_2 + nx_1}{m+n} \][/tex]
[tex]\[ y = \frac{my_2 + ny_1}{m+n} \][/tex]
#### (a) [tex]\( A = (1,1) \)[/tex] and [tex]\( B = (4,4) \)[/tex]; Ratio [tex]\( = 1 : 2 \)[/tex]
[tex]\[ m = 1, n = 2 \][/tex]
[tex]\[ x = \frac{1 \cdot 4 + 2 \cdot 1}{1+2} = \frac{4 + 2}{3} = 2.0 \][/tex]
[tex]\[ y = \frac{1 \cdot 4 + 2 \cdot 1}{1+2} = \frac{4 + 2}{3} = 2.0 \][/tex]
So 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]\[ m = 2, n = 1 \][/tex]
[tex]\[ x = \frac{2 \cdot 8 + 1 \cdot 2}{2+1} = \frac{16 + 2}{3} = 6.0 \][/tex]
[tex]\[ y = \frac{2 \cdot 3 + 1 \cdot 6}{2+1} = \frac{6 + 6}{3} = 4.0 \][/tex]
So 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]\[ m = 3, n = 1 \][/tex]
[tex]\[ x = \frac{3 \cdot 12 + 1 \cdot 4}{3+1} = \frac{36 + 4}{4} = 10.0 \][/tex]
[tex]\[ y = \frac{3 \cdot 1 + 1 \cdot 5}{3+1} = \frac{3 + 5}{4} = 2.0 \][/tex]
So 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]\[ m = 3, n = 2 \][/tex]
[tex]\[ x = \frac{3 \cdot 12 + 2 \cdot 2}{3+2} = \frac{36 + 4}{5} = 8.0 \][/tex]
[tex]\[ y = \frac{3 \cdot 1 + 2 \cdot 6}{3+2} = \frac{3 + 12}{5} = 3.0 \][/tex]
So 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]\[ m = 3, n = 2 \][/tex]
[tex]\[ x = \frac{3 \cdot 3 + 2 \cdot (-7)}{3+2} = \frac{9 - 14}{5} = -1.0 \][/tex]
[tex]\[ y = \frac{3 \cdot 0 + 2 \cdot 5}{3+2} = \frac{0 + 10}{5} = 2.0 \][/tex]
So 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]\[ m = 5, n = 1 \][/tex]
[tex]\[ x = \frac{5 \cdot 5 + 1 \cdot (-7)}{5+1} = \frac{25 - 7}{6} = 3.0 \][/tex]
[tex]\[ y = \frac{5 \cdot (-1) + 1 \cdot 5}{5+1} = \frac{-5 + 5}{6} = 0.0 \][/tex]
So the coordinates are [tex]\( (3.0, 0.0) \)[/tex].
### Part 2: Finding the ratio in which the point P divides AB internally
Given points [tex]\( A(x_1, y_1) \)[/tex], [tex]\( B(x_2, y_2) \)[/tex], and [tex]\( P(x, y) \)[/tex], the ratio [tex]\( m:n \)[/tex] in which P divides AB can be calculated using:
[tex]\[ m : n = \frac{x_2 - x}{x - x_1} \][/tex]
#### (a) [tex]\( A = (1,1) \)[/tex], [tex]\( B = (4,4) \)[/tex], and [tex]\( P = (2,2) \)[/tex]
[tex]\[ \frac{x_2 - x}{x - x_1} = \frac{4 - 2}{2 - 1} = \frac{2}{1} = 2 \][/tex]
So the ratio is [tex]\( 2 : 1 \)[/tex].
#### (b) [tex]\( A = (2,6) \)[/tex], [tex]\( B = (8,3) \)[/tex], and [tex]\( P = (6,4) \)[/tex]
[tex]\[ \frac{x_2 - x}{x - x_1} = \frac{8 - 6}{6 - 2} = \frac{2}{4} = 0.5 \][/tex]
So the ratio is [tex]\( 0.5 : 1 \)[/tex].
#### (c) [tex]\( A = (-3,3) \)[/tex], [tex]\( B = (5,-1) \)[/tex], and [tex]\( P = (3,0) \)[/tex]
[tex]\[ \frac{x_2 - x}{x - x_1} = \frac{5 - 3}{3 - (-3)} = \frac{2}{6} \approx 0.33 \][/tex]
So the ratio is [tex]\( 0.33 : 1 \)[/tex].
#### (d) [tex]\( A = (-3,3) \)[/tex], [tex]\( B = (5,-1) \)[/tex], and [tex]\( P = (-1,2) \)[/tex]
[tex]\[ \frac{x_2 - x}{x - x_1} = \frac{5 - (-1)}{-1 - (-3)} = \frac{6}{2} = 3 \][/tex]
So the ratio is [tex]\( 3 : 1 \)[/tex].
#### (e) [tex]\( A = (2,6) \)[/tex], [tex]\( B = (12,1) \)[/tex], and [tex]\( P = (10,2) \)[/tex]
[tex]\[ \frac{x_2 - x}{x - x_1} = \frac{12 - 10}{10 - 2} = \frac{2}{8} = 0.25 \][/tex]
So the ratio is [tex]\( 0.25 : 1 \)[/tex].
#### (f) [tex]\( A = (2,6) \)[/tex], [tex]\( B = (14,0) \)[/tex], and [tex]\( P = (12,1) \)[/tex]
[tex]\[ \frac{x_2 - x}{x - x_1} = \frac{14 - 12}{12 - 2} = \frac{2}{10} = 0.2 \][/tex]
So the ratio is [tex]\( 0.2 : 1 \)[/tex].
