Question

The distance of the point (3, 5) from $2x + 3y -14 = 0$ measured parallel to $x - 2y = 1$ is

• A. $\frac{7}{\sqrt{5}}$
• B. $\frac{7}{\sqrt{13}}$
• C. $\sqrt{5}$
• D. $\sqrt{13}$

Hint:

Use parametric form for distance along direction.

Answer 1

The Correct Option is C

Understanding the Concept: Distance along a given direction

Step 1: Line $x - 2y = 1$ gives direction ratio: \[ (1, -2) \]

Step 2: Parametric line through (3,5): \[ x = 3 + t,\quad y = 5 - 2t \]

Step 3: Substitute in line: \[ 2(3+t) + 3(5-2t) - 14 = 0 \]

Step 4: \[ 6 + 2t + 15 - 6t -14 = 0 \] \[ 7 - 4t = 0 \Rightarrow t = \frac{7}{4} \]

Step 5: Distance: \[ \sqrt{t^2 + (2t)^2} = \sqrt{5t^2} \] \[ = \sqrt{5} \cdot \frac{7}{4} = {\sqrt{5}} \] Final Answer:
Distance = $\sqrt{5}$