Question

The line \( AB \) passes through the point \( P(-4, 3) \) and the portion of the line intercepted between the axes is divided internally in the ratio 5:3 by the point \( P \). Given that the point \( A \) lies on the x-axis and \( B \) lies on the y-axis, then the x-intercept of the line is

• A. \( \frac{32}{3} \)
• B. \( \frac{32}{5} \)
• C. \( \frac{24}{5} \)
• D. \( \frac{24}{3} \)

Hint:

To find the intercepts of a line given by two points, use the section formula to divide the segment and determine the equation of the line. Then, solve for the intercepts.

Answer 1

The Correct Option is A

Step 1: Use the section formula.
The coordinates of point \( P(-4, 3) \) divide the line segment \( AB \) in the ratio 5:3. Let the coordinates of \( A \) be \( (x_1, 0) \) (since \( A \) lies on the x-axis) and the coordinates of \( B \) be \( (0, y_1) \) (since \( B \) lies on the y-axis).
Using the section formula, the coordinates of \( P(x, y) \), dividing \( AB \) in the ratio \( m:n \), are given by:
\[ x = \frac{m x_2 + n x_1}{m + n}, \quad y = \frac{m y_2 + n y_1}{m + n} \]
In our case, \( P(-4, 3) \) divides \( AB \) in the ratio 5:3, so \( m = 5 \), \( n = 3 \), and the coordinates of \( A \) are \( (x_1, 0) \), \( B(0, y_1) \).

Step 2: Set up equations.
From the section formula, we know:
\[ -4 = \frac{5(0) + 3(x_1)}{5 + 3} = \frac{3x_1}{8} \]
Solving for \( x_1 \):
\[ x_1 = \frac{-4 \times 8}{3} = \frac{-32}{3} \]

Step 3: Find the equation of the line.
The slope \( m \) of the line passing through the points \( A(x_1, 0) \) and \( B(0, y_1) \) is given by:
\[ m = \frac{y_1 - 0}{0 - x_1} = \frac{y_1}{-x_1} \]
Substitute \( x_1 = \frac{-32}{3} \):
\[ m = \frac{y_1}{\frac{32}{3}} = \frac{3y_1}{32} \]

Step 4: Use the equation of the line.
The equation of the line is in the form \( y = mx + c \), where \( c = y_1 \). Since the line passes through point \( P(-4, 3) \), substitute \( P(-4, 3) \) into the equation of the line to find \( y_1 \):
\[ 3 = \frac{3y_1}{32}(-4) + y_1 \]
Simplify:
\[ 3 = \frac{-12y_1}{32} + y_1 \]
Multiply through by 32:
\[ 96 = -12y_1 + 32y_1 \] Simplifying:
\[ 96 = 20y_1 \]
Solving for \( y_1 \):
\[ y_1 = \frac{96}{20} = \frac{24}{5} \]

Step 5: Find the x-intercept.
Now that we know \( x_1 = \frac{-32}{3} \), the x-intercept of the line is \( \boxed{\frac{32}{3}} \), which corresponds to option (A).