Question

The points $ (2,3) $ and $ \left(-4, \frac{4}{3}\right) $ lie on opposite sides of the line $$ L = 5x - 6y + k = 0, $$ and $ k $ is an integer. If the points $ (1, 2) $ and $ (4, 5) $ lie on the same side of the line, then the perpendicular distance from the origin to the line $ L = 0 $ is?

• A. \( \frac{7}{\sqrt{61}} \)
• B. \( \frac{9}{\sqrt{61}} \)
• C. \( \frac{10}{\sqrt{61}} \)
• D. \( \frac{11}{\sqrt{61}} \)

Hint:

Use sign of line equation at points to determine side, and formula for perpendicular distance from point to line.

Answer 1

The Correct Option is D

1. Substitute points into \( L \): \[ L(2,3) = 5(2) - 6(3) + k = 10 - 18 + k = k - 8 \] \[ L\left(-4, \frac{4}{3}\right) = 5(-4) - 6 \left(\frac{4}{3}\right) + k = -20 - 8 + k = k - 28 \] Since these points lie on opposite sides, \( L(2,3) \cdot L(-4, \frac{4}{3})<0 \): \[ (k - 8)(k - 28)<0 \Rightarrow 8<k<28 \] 2. Substitute \( (1,2) \) and \( (4,5) \): \[ L(1,2) = 5 - 12 + k = k - 7 \] \[ L(4,5) = 20 - 30 + k = k - 10 \] Same side means \( (k - 7)(k - 10)>0 \), so: \[ k<7 \quad \text{or} \quad k>10 \] 3. Combine the two conditions: \[ 8<k<28 \quad \text{and} \quad (k<7 \text{ or } k>10) \Rightarrow k>10 \] Since \( k \) is integer, smallest integer \( k = 11 \). 4. Distance from origin to line: \[ d = \frac{|k|}{\sqrt{5^2 + (-6)^2}} = \frac{11}{\sqrt{25 + 36}} = \frac{11}{\sqrt{61}} \]