Question

The ratio in which the X-axis divide's the line segment joining the points (4, 6) and (3.-8) is

• A. 1:2
• B. 2:3
• C. 3:4
• D. 4:5

Answer 1

The Correct Option is C

Step 1: Recall the section formula.

If a point divides a line segment joining two points \( (x_1, y_1) \) and \( (x_2, y_2) \) in the ratio \( m:n \), then the coordinates of the dividing point are given by:

\[ \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right). \]

Here, the dividing point lies on the X-axis, so its \( y \)-coordinate is \( 0 \).

Step 2: Use the condition for the X-axis.

The \( y \)-coordinate of the dividing point is \( 0 \). Using the section formula for the \( y \)-coordinate:

\[ \frac{my_2 + ny_1}{m+n} = 0. \]

Substitute \( y_1 = 6 \) and \( y_2 = -8 \):

\[ \frac{m(-8) + n(6)}{m+n} = 0. \]

Simplify:

\[ -8m + 6n = 0. \]

Rearrange:

\[ 8m = 6n \implies \frac{m}{n} = \frac{6}{8} = \frac{3}{4}. \]

Step 3: Interpret the result.

The ratio \( m:n \) is \( 3:4 \).

Final Answer: The ratio in which the X-axis divides the line segment is \( \mathbf{3:4} \), which corresponds to option \( \mathbf{(3)} \).