Question

In what ratio the segment joining \( (-1, -12) \) and \( (3, 4) \) divided by x-axis?

• A. \( 1:3 \)
• B. \( 2:1 \)
• C. \( 3:1 \)
• D. \( 5:1 \)

Hint:

A shortcut: The ratio in which the x-axis divides the line segment joining \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by \( -y_1 : y_2 \). Here, ratio = \( -(-12) : 4 = 12 : 4 = 3 : 1 \).

Answer 1

The Correct Option is C

Step 1: Let the two given points be \( A = (-1, -12) \) and \( B = (3, 4) \). Let the point P divide the line segment AB in the ratio \( m:n \). Since the point P lies on the x-axis, its y-coordinate must be 0. Let the coordinates of P be \( (x, 0) \).

Step 2: Using the section formula, the coordinates of the point P dividing the line segment joining \( (x_1, y_1) \) and \( (x_2, y_2) \) in the ratio \( m:n \) are given by: \[ P(x, y) = \left( \frac{m x_2 + n x_1}{m + n}, \frac{m y_2 + n y_1}{m + n} \right) \] Here, \( (x_1, y_1) = (-1, -12) \) and \( (x_2, y_2) = (3, 4) \).

Step 3: Substituting the coordinates and the fact that the y-coordinate of P is 0: \[ y = \frac{m y_2 + n y_1}{m + n} = 0 \] \[ \frac{m(4) + n(-12)}{m + n} = 0 \] \[ \frac{4m - 12n}{m + n} = 0 \]

Step 4: For the fraction to be zero, the numerator must be zero (assuming \( m+n \neq 0 \), which is true for a ratio). \[ 4m - 12n = 0 \] \[ 4m = 12n \] \[ \frac{m}{n} = \frac{12}{4} \] \[ \frac{m}{n} = \frac{3}{1} \] Therefore, the ratio \( m:n \) is \( 3:1 \).