Question

In what ratio does the point P(2, −5) divide the line segment joining A(−3, 5) and B(4, −9)

Answer 1

A line divides internally in the ratio m : n

The point P = `(("m"x_2 + "n"x_1)/("m" + "n"), ("m"y_2 + "n"y_1)/("m" + "n"))`

x₁ = −3, x₂ = 4, y₁ = 5, y₂ = −9

(2, −5) = `((4"m" - 3"n")/("m" + "n"), (-9"m" + 5"n")/("m" + "n"))`

`(4"m" - 3"n")/("m" + "n")`

= 2

4m – 3n = 2m + 2n

4m – 2m = 3n + 2n

2m = 5n

`"m"/"n" = 5/2`

m : n = 5 : 2

The ratio is 5 : 2.

and

`(-9"m" + 5"n")/("m" + "n")`

= −5

−9m + 5n = −5(m + n)

−9m + 5n = −5m – 5n

−9m + 5m = −5n – 5n

−4m = −10

`"m"/"n" = 10/4`

⇒ `"m"/"n" = 5/2`

m : n = 5 : 2