Question

Find the ratio in which the lines segment joining A(1,-5) & B(-4,5) is divided by the x-axis. Also find the corrdinates of the point of the division.​

Answer 1

Step-by-step explanation:

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

Given that

A line segment AB having coordinates of A as (1, - 5) and coordinates of B as (- 4, 5).

Let assume that x - axis divides the line segment joining the points A (1, - 5) and B (- 4, 5) in the ratio k : 1 at C.

Let assume that coordinates of C be (x, 0).

We know,

Section formula :-

Let P(x₁, y₁) and Q(x₂, y₂) be two points in the coordinate plane and R(x, y) be the point which divides PQ internally in the ratio m₁ : m₂. Then, the coordinates of R will be:

$\sf\implies \boxed{\tt{ R = \bigg(\dfrac{m_{1}x_{2}+m_{2}x_{1}}{m_{1}+m_{2}}, \dfrac{m_{1}y_{2}+m_{2}y_{1}}{m_{1}+m_{2}}\bigg)}}$

So, on substituting the values, we get

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

On comparing y - coordinate on both sides, we get

$\rm \longmapsto\:\dfrac{5k - 5}{k + 1} = 0$

$\rm \longmapsto\:5k - 5 = 0$

$\rm \longmapsto\:5k = 5$

$\bf\implies \:k = 1$

Hence,

The x - axis divides the line segment joining the points A (1, - 5) and B (- 4, 5) in the ratio 1 : 1 at C.

Now, On comparing x - coordinate on both sides, we get

$\rm \longmapsto\:x = \dfrac{ - 4k + 1}{k + 1}$

On substituting the value of k, we get

$\rm \longmapsto\:x = \dfrac{ - 4+ 1}{1 + 1}$

$\rm \longmapsto\:x \: = - \: \dfrac{3}{2}$

Hence,

The coordinates of point of intersection, C is

$\rm\implies \:\boxed{\tt{ Coordinates \: of \: C = \bigg( - \dfrac{3}{2}, \: 0 \bigg) }}$