Question

If the coordinates of the points A and B are (3, 3)  and  (7, 6),  then the length of the portion of the line AB intercepted between the axes is:

• A. \( \frac{5}{4} \)
• B. \( \frac{\sqrt{10}}{4} \)
• C. \( \frac{\sqrt{13}}{3} \)
• D. {None of these}

Hint:

To find the length of the portion of a line intercepted between the axes, first find the intercepts with the axes using the equation of the line, and then use the distance formula to calculate the length.

Answer 1

The Correct Option is A

We are given two points: \( A(3, 3) \) and \( B(7, 6) \). We need to find the length of the portion of the line AB intercepted between the axes. 
First, find the equation of the line passing through points \( A(3, 3) \) and \( B(7, 6) \). The slope of the line is: \[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{6 - 3}{7 - 3} = \frac{3}{4}. \] Now, use the point-slope form of the line equation: \[ y - y_1 = m(x - x_1). \] Using point \( A(3, 3) \): \[ y - 3 = \frac{3}{4}(x - 3). \] Simplifying: \[ y - 3 = \frac{3}{4}x - \frac{9}{4} \quad \Rightarrow \quad y = \frac{3}{4}x - \frac{9}{4} + 3 = \frac{3}{4}x + \frac{3}{4}. \] Thus, the equation of the line is: \[ y = \frac{3}{4}x + \frac{3}{4}. \] Next, find the intercepts of the line with the axes: - For the \( x \)-intercept, set \( y = 0 \): \[ 0 = \frac{3}{4}x + \frac{3}{4} \quad \Rightarrow \quad x = -1. \] Thus, the \( x \)-intercept is \( (-1, 0) \). - For the \( y \)-intercept, set \( x = 0 \): \[ y = \frac{3}{4}(0) + \frac{3}{4} = \frac{3}{4}. \] Thus, the \( y \)-intercept is \( (0, \frac{3}{4}) \). Now, use the distance formula to find the length of the segment between the intercepts \( (-1, 0) \) and \( (0, \frac{3}{4}) \): \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} = \sqrt{(0 - (-1))^2 + \left(\frac{3}{4} - 0\right)^2}. \] Simplifying: \[ d = \sqrt{1^2 + \left(\frac{3}{4}\right)^2} = \sqrt{1 + \frac{9}{16}} = \sqrt{\frac{16}{16} + \frac{9}{16}} = \sqrt{\frac{25}{16}} = \frac{5}{4}. \] Thus, the length of the portion of the line AB intercepted between the axes is \( \frac{5}{4} \).