Step-by-step explanation:
[tex]\large\underline{\sf{Solution-}}[/tex]
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:
[tex]\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)}}[/tex]
So, on substituting the values, we get
[tex]\rm \longmapsto\:(x,0) = \bigg(\dfrac{ - 4k + 1}{k + 1}, \: \dfrac{5k - 5}{k + 1} \bigg) [/tex]
On comparing y - coordinate on both sides, we get
[tex]\rm \longmapsto\:\dfrac{5k - 5}{k + 1} = 0[/tex]
[tex]\rm \longmapsto\:5k - 5 = 0[/tex]
[tex]\rm \longmapsto\:5k = 5[/tex]
[tex]\bf\implies \:k = 1[/tex]
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
[tex]\rm \longmapsto\:x = \dfrac{ - 4k + 1}{k + 1} [/tex]
On substituting the value of k, we get
[tex]\rm \longmapsto\:x = \dfrac{ - 4+ 1}{1 + 1} [/tex]
[tex]\rm \longmapsto\:x \: = - \: \dfrac{3}{2} [/tex]
Hence,
The coordinates of point of intersection, C is
[tex]\rm\implies \:\boxed{\tt{ Coordinates \: of \: C = \bigg( - \dfrac{3}{2}, \: 0 \bigg) }}[/tex]
