P divides $¯ ¯¯¯¯¯¯ ¯ A B$ from A's side in the ratio $2 : 1.$ If A $(3, 8)$ and P $(1, 12)$ find the co-ordinate of B.

(10, 14)

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(0, - 14)

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(0, 14)

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(0, 4)

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Solution

The correct option is C $(0, 14)$
Let $A \equiv (x, y), B \equiv (x_{2}, y_{2}), P = (x, y)$ and let $P$ divide
$\to A B$ in ratio $m . n$ .

Clearly

$x = \frac{n x_{1} + m x_{2}}{n + m}$ , $y = \frac{n y_{1} + m y_{2}}{n + m}$

Here, $x_{1} = 3, y_{1} = 8, x = 1, y = 12$ and $m = 2, n = 1$

Thus $x_{2} = \frac{(m + n) x - n x}{m}, y_{2} = \frac{(m + n) y - n y}{m}$

$x_{2} = \frac{(2 + 1) (1) - 1 (3)}{2}, y_{2} = \frac{(3) (12) - (1) (8)}{2}$

$x_{2} = 0, y_{2} = \frac{36 - 8}{2} - \frac{28}{2} = 14$

$∴ B \equiv (0, 14)$

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