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:

To find the division ratio by $x$-axis directly:
\[ \text{Ratio } = -\frac{y_1}{y_2} \]
Here, $y_1 = 5$ and $y_2 = -1$:
\[ \text{Ratio } = -\frac{5}{-1} = \frac{5}{1} = 5 : 1 \]
To find the division ratio by $y$-axis directly:
\[ \text{Ratio } = -\frac{x_1}{x_2} \]
These formulas are extremely handy for verification!

Answer 1

Step 1: Understanding the Question:
The topic of this question is Coordinate Geometry, specifically the Section Formula.
We have a line segment joining two given points, and the $x$-axis divides this segment.
We need to find the ratio of this division and the coordinates of the intersection point on the $x$-axis.

Step 2: Key Formula or Approach:
Let the ratio in which the $x$-axis divides the segment be $k : 1$.
By Section Formula, the coordinates of the point $P(x, y)$ that divides the segment joining $A(x_1, y_1)$ and $B(x_2, y_2)$ in the ratio $k : 1$ are:
\[ P(x, y) = \left( \frac{kx_2 + x_1}{k + 1}, \frac{ky_2 + y_1}{k + 1} \right) \]
Since the dividing point $P$ lies on the $x$-axis, its $y$-coordinate must be 0 ($y = 0$).
We can use this condition to solve for $k$ first, and then find the $x$-coordinate.

Step 3: Detailed Explanation:

• Let the given points be $A(x_1, y_1) = (-6, 5)$ and $B(x_2, y_2) = (-4, -1)$.
Let the $x$-axis divide the line segment $AB$ at point $P(x, 0)$ in the ratio $k : 1$.

• Write down the Section Formula for the $y$-coordinate of point $P$:
\[ y = \frac{ky_2 + y_1}{k + 1} \]

• Since $P$ is on the $x$-axis, set $y = 0$:
\[ 0 = \frac{k(-1) + 5}{k + 1} \]
Multiply both sides by $(k + 1)$:
\[ -k + 5 = 0 \]
\[ k = 5 \]
So, the ratio in which the $x$-axis divides the line segment is $5 : 1$.

• Now, write down the Section Formula for the $x$-coordinate of point $P$:
\[ x = \frac{kx_2 + x_1}{k + 1} \]

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

• Thus, the coordinates of the point of intersection are:
\[ P(x, y) = \left( -\frac{13}{3}, 0 \right) \]


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