Question

The ratio in which the x-axis divides the line segment joining the points A\(-8, 4\) and B\(-6, -2\) is :

• A. 5 : 1
• B. 3 : 1
• C. 2 : 1
• D. 1 : 2

Hint:

A useful shortcut to find the ratio in which the \(x\)-axis divides a line segment joining \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[ \text{Ratio} = -\frac{y_1}{y_2} \]
Substitute the given \(y\)-coordinates:
\[ \text{Ratio} = -\frac{4}{-2} = \frac{4}{2} = 2 = 2 : 1 \]
Similarly, the \(y\)-axis divides a line segment in the ratio \(-\frac{x_1}{x_2}\).

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
We need to determine the ratio in which the \(x\)-axis divides the line segment joining the points \(A(-8, 4)\) and \(B(-6, -2)\).

Step 2: Key Formula or Approach:
According to the section formula, if a point \(P\) divides the line segment joining \(A(x_1, y_1)\) and \(B(x_2, y_2)\) in the ratio \(k : 1\), the coordinates of \(P\) are:
\[ P(x, y) = \left( \frac{kx_2 + x_1}{k + 1}, \frac{ky_2 + y_1}{k + 1} \right) \]
Since the division is done by the \(x\)-axis, the \(y\)-coordinate of the point of division \(P\) must be 0 (since any point on the \(x\)-axis is of the form \((x, 0)\)).

Step 3: Detailed Explanation:
1. Let the required ratio be \(k : 1\) in which the \(x\)-axis divides the line segment \(AB\).
2. The given coordinates are:
\[ A(x_1, y_1) = (-8, 4) \]
\[ B(x_2, y_2) = (-6, -2) \]
3. The \(y\)-coordinate of the point of intersection \(P\) on the \(x\)-axis is given by:
\[ y = \frac{ky_2 + y_1}{k + 1} \]
4. Since the point lies on the \(x\)-axis, set \(y = 0\):
\[ \frac{k(-2) + 4}{k + 1} = 0 \]
5. Solve the equation for \(k\):
\[ -2k + 4 = 0 \]
\[ 2k = 4 \implies k = 2 \]
6. Since \(k = 2\), the ratio is \(k : 1\), which simplifies to \(2 : 1\).

Step 4: Final Answer:
The \(x\)-axis divides the line segment in the ratio \(2 : 1\), which corresponds to option (C).