Question

Find the ratio in which the line joining (4, -3) and ( 2, 5) is divided internally by X axis​

Answer 1

$\large\underline{\sf{Solution-}}$

Let the given coordinates be represented by A ( 4, - 3 ) and B ( 2, 5 ).

Let further assume that the x - axis divides the line segment AB in the ratio k : 1 and intersects AB at C.

Let coordinates of C be ( x, 0 ).

So, We know Section Formula :-

Let us consider a line segment joining the points A (x₁ , y₁ ) and B (x₂ , y₂) and Let C (x, y) be any point on AB which divides AB internally in the ratio m : n, then coordinates of C is

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

So, using this

Here,

• x₁ = 4

• x₂ = 2

• y₁ = - 3

• y₂ = 5

• m = k

• n = 1

On substituting these values in above formula we get

$\rm :\longmapsto\:( x, 0 ) = \bigg(\dfrac{5k + 4}{k + 1} , \: \dfrac{2k - 3}{k + 1} \bigg)$

On comparing y - coordinates on both sides, we get

$\rm :\longmapsto\:\dfrac{2k - 3}{k + 1} = 0$

$\rm :\longmapsto\:2k - 3 = 0$

$\rm :\longmapsto\:2k = 3$

$\bf\implies \:k = \dfrac{3}{2}$

Hence,

• The required ratio is 3 : 2

Additional Information :-

1. Distance Formula:-

Let us consider a line segment joining the points A (x₁ , y₁ ) and B (x₂ , y₂) , then distance between A and B is

${\underline{\boxed{\rm{\quad Distance = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \quad}}}}$

2. Midpoint Formula :-

Let us consider a line segment joining the points A (x₁ , y₁ ) and B (x₂ , y₂) and Let C (x, y) be mid - point on AB, then coordinates of C is

$\green{\boxed{ \bf{ ( x, y ) = \bigg(\dfrac{x_1 + x_2}{2} , \: \dfrac{y_1 + y_2}{2} \bigg)}}}$