Question 5

Line Segment Joining: The line segment joining two points A and B is the straight path between them. Any point that lies on this segment divides it in some ratio $m:n$.

Point of Division: When a line (here the x-axis) cuts through a line segment, the intersection point divides the segment in a specific ratio. Any point on the x-axis has y-coordinate = 0, which we use to find the ratio.

Section Formula: If a point $\text{P}(x, y)$ divides the line segment joining $\text{A}(x_1, y_1)$ and $\text{B}(x_2, y_2)$ in the ratio $m:n$ internally, then:

$x = \frac{mx_2 + nx_1}{m + n}, \quad y = \frac{my_2 + ny_1}{m + n}$

We let the ratio be $k:1$, set $y = 0$ (since the x-axis has y = 0), and solve for $k$.

We will use the section formula to solve this problem.

Step 1 — Find the ratio

Let the x-axis divide the line segment in the ratio $k:1$. The y-coordinate of any point on the x-axis is 0. We use the section formula for the y-coordinate.

$y = \frac{k y_2 + 1 y_1}{k+1}$

$0 = \frac{k(5) + 1(-5)}{k+1}$

$0 = \frac{5k - 5}{k+1}$

$0 \times (k+1) = 5k - 5$

$0 = 5k - 5$

$5k = 5$

$k = 1$

$\boxed{\text{Ratio is } 1:1}$

Step 2 — Find the coordinates

Now we know the ratio is 1:1. We can find the x-coordinate of the point of division. We use the section formula for the x-coordinate.

$x = \frac{k x_2 + 1 x_1}{k+1}$

$x = \frac{1(-4) + 1(1)}{1+1}$

$x = \frac{-4 + 1}{2}$

$x = \frac{-3}{2}$

$\boxed{\text{Coordinates are } \left(-\frac{3}{2}, 0\right)}$

Answer

(i) The ratio is 1:1. (ii) The coordinates of the point of division are $\left(-\frac{3}{2}, 0\right)$.