Question

Find the ratio in which the x-axis divides the line segment joining the points ($-$6, 5) and ($-$4, $-$1). Also, find the point of intersection.

Hint:

For any line segment divided by the x-axis, the ratio $k:1$ is always given by:
\[ k = -\frac{y_1}{y_2} \]
Substituting $y_1 = 5$ and $y_2 = -1$ gives:
\[ k = -\frac{5}{-1} = 5 \]
This is a robust shortcut for calculating division ratios!

Answer 1

Step 1: Understanding the Question:
The question asks us to find the ratio in which the horizontal coordinate axis (x-axis) divides the straight line segment joining two given points, $A(-6, 5)$ and $B(-4, -1)$.
We also need to determine the exact coordinates of this point of intersection on the x-axis.

Step 2: Key Formula or Approach:
1. Let the ratio in which the x-axis divides the line segment joining $A(x_1, y_1)$ and $B(x_2, y_2)$ be $k : 1$.
2. According to the Section Formula, the coordinates of the dividing point $P(x, y)$ are given by:
\[ P(x, y) = \left( \frac{k x_2 + x_1}{k + 1}, \frac{k y_2 + y_1}{k + 1} \right) \]
3. Since the point of intersection lies on the x-axis, its y-coordinate must be zero ($y = 0$). We use this key condition to find $k$.

Step 3: Detailed Explanation:

• Identify the coordinates of the given points:
Let $(x_1, y_1) = (-6, 5)$ and $(x_2, y_2) = (-4, -1)$.

• Express the y-coordinate of the dividing point $P$ and set it to zero:
\[ y = \frac{k y_2 + y_1}{k + 1} = 0 \]
Substitute the values $y_1 = 5$ and $y_2 = -1$:
\[ \frac{k(-1) + 5}{k + 1} = 0 \]

• Solve for $k$:
\[ -k + 5 = 0 \implies k = 5 \]
So, the ratio in which the x-axis divides the segment is $5 : 1$.

• Calculate the x-coordinate of the intersection point $P$:
Substitute $k = 5$, $x_1 = -6$, and $x_2 = -4$ into the section formula for $x$:
\[ x = \frac{k x_2 + x_1}{k + 1} \]
\[ x = \frac{5(-4) + (-6)}{5 + 1} \]
\[ x = \frac{-20 - 6}{6} = \frac{-26}{6} = -\frac{13}{3} \]

• Write down the coordinates of the point of intersection:
The point is $P\left(-\frac{13}{3}, 0\right)$.


Step 4: Final Answer:
The x-axis divides the line segment in the ratio $5 : 1$, and the point of intersection is $\left(-\frac{13}{3}, 0\right)$.