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: Understand what is being asked.
We need the ratio in which the $x$-axis divides the segment joining $A(-8,4)$ and $B(-6,-2)$. Instead of setting the $y$-coordinate of the section formula to zero, let us use the perpendicular distances of $A$ and $B$ from the $x$-axis directly.

Step 2: Note the key idea.
When a point on the $x$-axis divides a segment joining two points on opposite sides of the axis, the two triangles formed by dropping perpendiculars from $A$ and $B$ to the axis are similar (both are right angled, and they have equal vertically opposite angles at the point of division). So the ratio in which the axis divides the segment equals the ratio of the perpendicular distances of $A$ and $B$ from the axis.

Step 3: Find the perpendicular distances.
The perpendicular distance of a point from the $x$-axis is just the size of its $y$-coordinate, ignoring the sign.
For $A(-8,4)$, the distance from the $x$-axis is $4$.
For $B(-6,-2)$, the distance from the $x$-axis is $2$.

Step 4: Write the ratio.
Since $A$ and $B$ lie on opposite sides of the axis ($A$ above, $B$ below, shown by their $y$-coordinates having opposite signs), the point of division lies between them, and the ratio in which it divides $AB$ (from $A$'s side to $B$'s side) equals:
\[ \text{Ratio} = 4 : 2 = 2 : 1 \]
Final Answer:
The $x$-axis divides the segment in the ratio $2:1$, matching option (C).
\[ \boxed{2:1} \]