Question

From the following options, the nearest line to the origin is ....

• A. \(3x - 4y + 4 = 0\)
• B. \(2x - 3y = 5\)
• C. \(4x - 3y + 12 = 0\)
• D. \(5x - 2y = 3\)

Hint:

To find the nearest line to origin, compute: \[ \frac{|c|}{\sqrt{a^2+b^2}} \] for each Option and choose the smallest value.

Answer 1

The Correct Option is D

Concept:
Distance of a line \[ ax+by+c=0 \] from the origin is: \[ d=\frac{|c|}{\sqrt{a^2+b^2}} \] ip

Step 1: Find the distance for each line.
For (A): \[ 3x-4y+4=0 \] \[ d_A=\frac{|4|}{\sqrt{3^2+(-4)^2}}=\frac{4}{5} \] For (B): \[ 2x-3y-5=0 \] \[ d_B=\frac{|{-5}|}{\sqrt{2^2+(-3)^2}}=\frac{5}{\sqrt{13}} \] For (C): \[ 4x-3y+12=0 \] \[ d_C=\frac{12}{\sqrt{4^2+(-3)^2}}=\frac{12}{5} \] For (D): \[ 5x-2y-3=0 \] \[ d_D=\frac{3}{\sqrt{5^2+(-2)^2}}=\frac{3}{\sqrt{29}} \] ip

Step 2: Compare the distances.
Approximate values are: \[ d_A=0.8,\quad d_B\approx1.39,\quad d_C=2.4,\quad d_D\approx0.557 \] The smallest distance is for option (D). ip Hence, the correct answer is:
\[ \boxed{(D)\ 5x-2y=3} \]