Question

Determine the ratio in which the point M (k, 1) divides the line segment joining the points A (7, −2) and B (−5, 6). Also, find the value of K.

Answer 1

Given: A(7, −2), B(−5, 6), and M(k, 1)

Let the internal ratio AP : PB = λ : 1  .....(We write λ for the unknown)

Using the section formula:

P = `(((n xx x_1) + (m xx x_2)) / (m + n)), (((n xx y_1) + (m xx y_2)) / (m + n))`

x-coordinate of M:

k = `((1 xx x_1) + ("λ" xx x_2)) / ("λ" + 1)`

= `((1 xx 7) + "λ"(−5)) / ("λ" + 1)`

y-coordinate of M:

1 = `((1 xx y_1) + ("λ" xx y_2)) / ("λ" + 1)`

= `((1 xx (−2)) + ("λ" xx 6)) / ("λ" + 1)`

Solve for the ratio λ using the y-equation:

1 = `((1 xx (−2)) + ("λ" xx 6)) / ("λ" + 1)`

1(λ + 1) = −2 + 6λ

λ + 1 = −2 + 6λ

1 + 2 = 6λ − λ

3 = 5λ

∴ λ = `3 / 5`

So AP : PB = `3 / 5` : 1 = 3 : 5  .....(This is the ratio in which M divides AB)

Find k (the x-coordinate) by plugging λ = `3 / 5` into the x-equation:

k = `(7 + (3/5)(−5)) / (3/5 + 1)`

k = `(7 + (-15/5)) / (8/5)`

k = `(7 + (-3)) / (8/5)`

k = `4 / (8/5)`

k = `(4 xx 5) / 8`

k = `20 / 8`

∴ k = `5 / 2`

Hence, the final answers are: ratio AP : PB = 3 : 5, and K (the x‑coordinate of M) = `5 / 2`.