Question

If \( p \) is the length of the perpendicular from the origin to the line whose intercepts with the coordinate axes are \( \frac{1}{3} \) and \( \frac{1}{4} \), then the value of \( p \) is:

• A. \( \frac{3}{4} \)
• B. \( \frac{1}{12} \)
• C. \( 5 \)
• D. \( 12 \)
• E. \( \frac{1}{5} \)

Hint:

An alternative relation for the length of the perpendicular \( p \) from the origin to a line with intercepts \( a \) and \( b \) is: \[ \frac{1}{p^2} = \frac{1}{a^2} + \frac{1}{b^2} \] Plugging in the values: \( \frac{1}{p^2} = 3^2 + 4^2 = 25 \), so \( p = 1/5 \).

Answer 1

Answer: (E)

Concept: The equation of a straight line in intercept form is given by: \[ \frac{x}{a} + \frac{y}{b} = 1 \] where \( a \) and \( b \) are the x-intercept and y-intercept respectively. The length of the perpendicular \( p \) from the origin \( (0,0) \) to the line \( Ax + By + C = 0 \) is calculated using: \[ p = \frac{|C|}{\sqrt{A^2 + B^2}} \]

Step 1: Form the equation of the line.
Given x-intercept \( a = \frac{1}{3} \) and y-intercept \( b = \frac{1}{4} \). Substituting into the intercept form: \[ \frac{x}{1/3} + \frac{y}{1/4} = 1 \] \[ 3x + 4y = 1 \quad \Rightarrow \quad 3x + 4y - 1 = 0 \]

Step 2: Calculate the perpendicular distance from the origin.
Here, \( A = 3, B = 4, \) and \( C = -1 \). \[ p = \frac{|-1|}{\sqrt{3^2 + 4^2}} \] \[ p = \frac{1}{\sqrt{9 + 16}} \] \[ p = \frac{1}{\sqrt{25}} = \frac{1}{5} \]