Question

The length of perpendicular from the origin to the line \( \frac{x}{5} - \frac{y}{12} = 1 \) is:

• A. \( \frac{60}{13} \)
• B. \( \frac{5}{12} \)
• C. \( \frac{12}{5} \)
• D. \( \frac{13}{12} \)
• E. \( \frac{13}{60} \)

Hint:

Use the perpendicular distance formula from a point to a line \( Ax + By + C = 0 \) for quick calculation.

Answer 1

The Correct Option is A

The formula for the length of the perpendicular from the origin to a line \( Ax + By + C = 0 \) is: \[ {Length} = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}} \] The equation is \( \frac{x}{5} - \frac{y}{12} = 1 \), which can be rewritten as: \[ \frac{x}{5} - \frac{y}{12} - 1 = 0 \quad {where} \, A = \frac{1}{5}, B = -\frac{1}{12}, C = -1 \] Thus, the length of the perpendicular is: \[ \frac{|0 + 0 - 1|}{\sqrt{\left( \frac{1}{5} \right)^2 + \left( -\frac{1}{12} \right)^2}} = \frac{1}{\sqrt{\frac{1}{25} + \frac{1}{144}}} = \frac{1}{\sqrt{\frac{169}{3600}}} = \frac{60}{13} \]