Question

Find the coordinates of the point P which divides the join of A( - 2, 5 ) and B( 3, - 5 ) in the ratio 2 : 3.

Answer 1

$\underleftrightarrow{A( - 2 , 5 )\quad\quad P(X,Y)\quad\quad B(3,-5)}\\\underbrace{\quad \quad \mathsf{m_{1} \: or \:2}\quad \quad}\underbrace{\quad \quad \mathsf{ m_{2} \: or \:3}\quad \quad}$

Given points : - A( - 2 , 5 ) ; B( 3 , - 5 )

Ratio : - 2 : 3  = m₁ : m₂

Let the point P be ( X , Y ),

According to the section formula,

$\boxed{\mathsf{X} = \dfrac{\mathsf{m_{1}x_{2} + m_{2} x_{1}}}{\mathsf{m_{1} + m_{2} }}}$

and

$\boxed{\mathsf{Y} = \dfrac{\mathsf{m_{1}y_{2} + m_{2}y_{1}}}{\mathsf{m_{1} + m_{2} }}}$

In the given question, x₁ = - 2 , x₂ = 3 , y₁ = 5 , y₂ = - 5

         Applying formula,

$\implies\mathsf{X} = \dfrac{\mathsf{( 2 \times 3 )  + ( 3 \times - 2 ) }}{\mathsf{ 2 + 3}}\\\\\implies X = \dfrac{6-6}{5} \\\\\implies X = 0$

$\mathsf{Y} = \dfrac{\mathsf{( 2 \times - 5 ) + ( 3 \times 5 )}}{\mathsf{2+ 3 }}\\\\\implies Y = \dfrac{-10+15}{5}\\\\\implies Y = \dfrac{5}{5} \\\\\implies Y = 1$

Therefore the coordinates of the point P which divides the points A and B are ( X , Y ) i.e. ( 0 , 1 ).

Answer 2

Coordinates of A = (-2, 5)

Coordinates of B = (3, -5)

Ratio of P = 2 : 3

Find point p:

$\text {P} = \bigg( \dfrac{3(-2) +2(3)}{2+3} \ , \ \dfrac{3(5) +2(-5)}{2+3} \bigg)$

$\text {P} = \bigg( 0 \ , \ 1 \bigg)$

Answer: The coordinates of P is (0, 1)