Question

Required ra
SOLVED EXAMPLES
Example 1. Find the co-ordinates of the point P which divides the line segment joining
A(8,9) and B(-7, 4), internally in the ratio 2 : 3.
Clearly, y =
The co-ord​

Answer 1

We're given,

$\longrightarrow \sf{A(8, 9) = ({x}_{1}, {y}_{1})}$

$\longrightarrow \sf{P = (\alpha, \beta)}$

$\longrightarrow \sf{B(-7, 4) = ({x}_{2}, {y}_{2})}$

$\longrightarrow \sf{AP:PB = m:n = 2:3}$

$\sf{\\ \\}$

We know by section formula, when a point $\sf{P(\alpha, \beta)}$ intersects internally a line segment AB where $\sf{A({x}_{1},{y}_{1})}$ and $\sf{B({x}_{2},{y}_{2})}$ in the ratio m : n, then:-

$\quad \quad \longrightarrow \sf{\alpha = \dfrac{m{x}_{2} + n{x}_{1}}{m + n}}$

$\quad \quad \longrightarrow \sf{\beta = \dfrac{m{y}_{2} + n{y}_{1}}{m + n}}$

$\sf{\\ \\}$

So, putting the values,

For alpha,

$\longrightarrow \sf{\alpha = \dfrac{m{x}_{2} + n{x}_{1}}{m + n}}$

$\longrightarrow \sf{\alpha = \dfrac{(2 * -7) + (3 * 8)}{2 + 3}}$

$\longrightarrow \sf{\alpha = \dfrac{ - 14 + 24}{5}}$

$\longrightarrow \sf{\alpha = \dfrac{10}{5}}$

$\longrightarrow \underline{\sf{\alpha = 2}}$

$\sf{\\ \\}$

For beta,

$\longrightarrow \sf{\beta = \dfrac{m{y}_{2} + n{y}_{1}}{m + n}}$

$\longrightarrow \sf{\beta = \dfrac{(2 * 4) + (3 * 9)}{2 + 3}}$

$\longrightarrow \sf{\beta = \dfrac{ 8 + 27}{5}}$

$\longrightarrow \sf{\beta = \dfrac{35}{5}}$

$\longrightarrow \underline{\sf{\beta = 7}}$

$\sf{\\ \\}$

Hence the answer is,

$\longrightarrow{\underline{\underline{\sf{ P = (2,7)}}}}$