Question

In what ratio is the line joining the points (2, 3) and (4, 1) divides the line segment joining the points (1, 2) and (4, 3)?

Answer 1

Here, the equation of the line passing through A(2, 3) and B(4, 1):

Slope(m):

`m = (1 - 3)/(4 - 2)`

`m = (-2)/2`

∴ m = −1

Using the point–slope formula:

y − y₁ = m(x − x₁)

y − 3 = −1(x − 2)

y − 3 = −x + 2

x + y − 5 = 0

Let L(x, y) = x + y − 5 = 0 divide the line joining the points P(1, 2) and Q(4, 3) at the ratio:

Ratio = `(L(x_1, y_1))/(L(x_2, y_2))`

Evaluate points P(1, 2) and Q(4, 3):

⇒ P(1, 2) = L(1, 2)

= 1 + 2 − 5

= −2

⇒ Q(4, 3) = L(4, 3)

= 4 + 3 − 5

= 2

So,

Ratio = `-(-2)/2`

= `2/2`

∴ Ratio = `1/1`

Hence, the line divides the segment in the ratio 1 : 1.