Find the ratio in which P(4, m) divides the line segment joining the points A(2, 3) and B(6, -3). Hence find m.

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Solution

Let P divides line segment AB in the ratio k : 1.

Coordinates of P
$P = (\frac{m_{1} x_{2} + m_{2} x_{1}}{m_{1} + m_{2}}, \frac{m_{1} y_{2} + m_{2} y_{1}}{m_{1} + m_{2}}) (4, m) = (\frac{k \times 6 + 1 \times 2}{k + 1}, \frac{k \times (- 3) + 1 \times 3}{k + 1})$
$(4, m)$ = $(\frac{6 k + 2}{k + 1}, \frac{- 3 k + 3}{k + 1})$
On comparing, we get
$(\frac{6 k + 2}{k + 1} = 4)$
$\Rightarrow 6 k + 2 = 4 + 4 k$
$\Rightarrow 6 k - 4 k = 4 - 2$
$\Rightarrow 2 k = 2$
$\Rightarrow k = 1$
Hence, P divides AB in the ratio 1 : 1.
From (i), $\frac{- 3 (1) + 3}{1 + 1} = m$
$\Rightarrow \frac{- 3 + 3}{2} = m$
$\Rightarrow m = 0$

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