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Section Formula

Find the coor...

Question

Find the coordinates of the point $P$ which divides line segment $Q R$ internally in the ratio $m : n$ in the following example:

$Q \equiv (- 5, 8), R \equiv (4, - 4)$ and m:n $=$ 2:1.

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Solution

$L e t t h e c o - o r d i n a t e s o f t h e p o i n t P b e P (x, y) w h i c h d i v i d e s t h e l i n e s e g m e n t j o i n i n g t h e p o i n t s Q (x_{1}, y_{1}) = (- 5, 8) & R (x_{2}, y_{2}) = (4, - 4) i n t h e r a t i o m : n = 2 : 1.$

$T h e n, b y t h e s e c t i o n f o r m u l a,$

$x = \frac{n x_{1} + m x_{2}}{m + n} = \frac{1 \times (- 5) + 2 \times 4}{2 + 1} = 1 a n d$

$y = \frac{n y_{1} + m y_{2}}{m + n} = \frac{1 \times (8) + 2 \times (- 4)}{2 + 1} = 0 ∴$

$P (x, y) = P (1, 0) .$

$A n s - P (x, y) = P (1, 0) .$

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