Question

The coordinates of the point P dividing the line segment joining the points

A(1, 3) and B(4, 6) in the ratio 2 : 1 are

(A) (2, 4) (B) (3, 5) (C) (4, 2) (D) (5, 3)​

Answer 1

Basic Concept Used :-

Section Formula :-

Let us consider a line segment joining the points A and B and let C (x, y) be any point on the line segment AB which divides AB in the ratio m : n internally, then coordinates of C is

$\sf \:( x, y)=\bigg(\dfrac{mx_2+nx_1}{m+n},\dfrac{my_2+ny_1}{m + n}\bigg)$

$\sf \: where \: coordinates \: are \: A \: (x_1,y_1) \: and \: B \: (x_2,y_2)$

Let's solve the problem now!!

Given Coordinates of line segment AB,

• Coordinates of A (1, 3)

and

• Coordinates of B (4, 6)

and

• P divides AB in the ratio 2 : 1

Let

• Coordinates of P be (x, y)

Then,

Coordinates of P is given by

$\rm :\longmapsto\:\sf \:( x, y)=\bigg(\dfrac{mx_2+nx_1}{m+n},\dfrac{my_2+ny_1}{m + n}\bigg)$

Here,

• • x₁ = 1

• • x₂ = 4

• • y₁ = 3

• • y₂ = 6

• • m = 2

• • n = 1

On substituting all these values in above formula, we get

$\rm :\longmapsto\:\sf \:( x, y)=\bigg(\dfrac{2 \times 4 + 1 \times 1}{2 + 1} ,\dfrac{2 \times 6 + 1 \times 3}{2 + 1} \bigg)$

$\rm :\longmapsto\:\sf \:( x, y)=\bigg(\dfrac{8 + 1}{3} ,\dfrac{12 + 3}{3} \bigg)$

$\rm :\longmapsto\:\sf \:( x, y)=\bigg(\dfrac{9}{3} ,\dfrac{15}{3} \bigg)$

$\rm :\longmapsto\:\sf \:( x, y)=\bigg(3, \: 5\bigg)$

$\boxed{ \bf \: Hence, \: Coordinates \: of \: P \: is \: (3, 5)}$

$\bf\implies \:Option \: (B) \: is \: correct$

Additional Information :-

1. Distance between two points is calculated by using the formula given below,

$\rm D = \sqrt{ {(x_{2} - x_{1}) }^{2} + {(y_{2} - y_{1})}^{2} }$

2. Area of triangle whose vertices are given is

$\begin{gathered} \sf Area \:of\:a\: triangle= \dfrac{1}{2} \begin{vmatrix} \sf x_1 - x_2 & \sf x_1 - x_3 \\\\\sf y_1 - y_2 & \sf y_1 - y_3 \end{vmatrix} \end{gathered}$

Or

$\sf \: A =\dfrac{1}{2} [x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)]$