Question

A point P(x, 7) divides a line segment joining the points A(−5, 4) and B(7, 9) in a certain ratio. Find the ratio and hence find the value of x.

Answer 1

Let A(x₁, y₁), B(x₂, y₂) and P(x, y) be the given points.

Here, x₁ = −5, y₁ = 4, x₂ = 7, y₂ = 9

By the section formula,

`P(x, y) = ((mx_2 + nx_1)/(m + n), (my_2 + ny_1)/(m + n))`

Using the known y-coordinate of P(y = 7), A(y₁ = 4), and B(y₂ = 9)

`y = (my_2 + ny_1)/(m + n)`

`7 = (m(9) + n(4))/(m + n)`

7(m + n) = 9m + 4n

7m + 7n = 9m + 4n

7n − 4n = 9m − 7m

3n = 2m

∴ `m/n = 3/2`

Thus, the ratio of m : n is 3 : 2.

Now, substitute the ratio m = 3 and n = 2 into the section formula for the x-coordinate:

`x = (mx_2 + nx_1)/(m + n)`

`x = (m(7) + n(-5))/(m + n)`

= `(3(7) + 2(-5))/(3 + 2)`

= `(21 - 10)/5`

∴ x = `11/5`

Hence, the point P divides the line segment AB in the ratio 3 : 2, and the value of x is `11/5`.