In what ratio does the line 3x plus 4y equals 7 divide the line segment joining the points 1 2 and -2 1?

Answer 1

First find the equation of the line containing the points (1, 2) and (-2, 1)

y - Y = m(x - X)

→ y - 2 = ((2 - 1)/(1 - -3))(x - 1)

→ y - 2 = (1/3)(x - 1)

→ 3y - 6 = x - 1

→ x = 3y - 5

This can now be substituted into the equation of the line 3x + 4y = 7 to find the point where they meet:

3x + 4y = 7

→ 3(3y - 5) + 4y = 7

→ 9y - 15 + 4y = 7

→ 13y = 22

→ y = 22/13

→ x = 3y - 5 = 3(22/13) - 5 = 1/13

So they meet at the point (1/13, 22/13)

Using Pythagoras, the length to each of the points (1, 2) and (-2, 1) from the point of meeting can be found, and thus their ratio:

To (1, 2):

length = √((1 - 1/13)² + (2 - 22/13)²)

= (√((13 - 1)² + (26 - 22)²))/13

= (√(12² + 4²))/13

= 4 × (√(3² + 1))/13

= 4 × (√10)/13

To (-2, 1):

length = √((-2 - 1/13)² + (1 - 22/13)²)

= (√((-26 - 1)² + (13 - 22)²))/13

= (√(27² + 9²))/13

= 9 × (√(3² + 1))/13

= 9 × (√10)/13

Giving the ratio of the lengths:

4 × (√10)/13 : 9 × (√10)/13

→ 4 : 9

Thus the line 3x + 4y = 7 divides the line segment joining (1, 2) to (-2, 1) in the ratio 4 : 9.