Find the ratio in which the point (−3,k) divides the join

Q) Find the ratio in which the point (−3, k) divides the join of A (−5, −4) and B (−2, 3). Also, find the value of k.

Ans: Let A (- 5,- 4), B (- 2,3), P (- 3, k).

Step 1: Using section formula for x coordinate:

$x = \frac{mx_2 + nx_1}{m+n}, \; y = \frac{my_2 + ny_1}{m+n}$

$-3 = \frac{m(-2) + n(-5)}{m+n}$

$-3(m+n) = -2m - 5n \;\;\Rightarrow\;\; m = 2n$

∴ m:n = 2: 1

Let’s consider, m = 2 p and n = p

Step 2: Using section formula for y coordinate:

$k = \frac{m(3) + n(-4)}{m+n}$

∴ $k = \frac{(2p)(3) + (p)(-4)}{2p + p}$

∴ $k = \frac{(6p - 4p)}{3p} = \frac{2p}{3p}$

∴ $k = \frac{2}{3}$

Therefore, value of k is $\frac{2}{3}$

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