Question

The length of the perpendicular from the point \( P(a,b) \) to the line \( \dfrac{x}{a} + \dfrac{y}{b} = 1 \) is

• A. \( \left| \dfrac{\sqrt{a^2+b^2}}{ab} \right| \) units
• B. \( \left| \dfrac{ab}{\sqrt{a^2+b^2}} \right| \) units
• C. \( \left| \dfrac{b^2}{\sqrt{a^2+b^2}} \right| \) units
• D. \( \left| \dfrac{a^2}{\sqrt{a^2+b^2}} \right| \) units

Hint:

Always convert a line into the form \( Ax+By+C=0 \) before applying the distance formula.

Answer 1

The Correct Option is B

Step 1: Write the line in general form.
Given line: \[ \frac{x}{a} + \frac{y}{b} - 1 = 0 \] Multiplying by \( ab \), \[ bx + ay - ab = 0 \] Step 2: Use the perpendicular distance formula.
Distance from point \( (x_1,y_1) \) to line \( Ax + By + C = 0 \) is \[ d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}} \] Step 3: Substitute values.
\[ d = \frac{|b(a) + a(b) - ab|}{\sqrt{b^2 + a^2}} = \frac{|ab|}{\sqrt{a^2+b^2}} \] Step 4: Conclusion.
The required perpendicular length is \[ \left| \frac{ab}{\sqrt{a^2+b^2}} \right| \]