Find the ratio in which the point P(x, 2) divides the join of A(12, 5) and B(4, −3).

Question:

Find the ratio in which the point P(x, 2) divides the join of A(12, 5) and B(4, −3). 

Solution:

Let k be the ratio in which the point P(x, 2) divides the line joining the points A(x1 = 12, y1 = 5) and B(x2 = 4, y2 = −3). Then

$x=\frac{k \times 4+12}{k+1} \quad$ and $\quad 2=\frac{k \times(-3)+5}{k+1}$

Now

$2=\frac{k \times(-3)+5}{k+1} \Rightarrow 2 k+2=-3 k+5 \Rightarrow k=\frac{3}{5}$

Hence, the required ratio is 3 : 5.

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