Step 1: Understanding the Concept:
The perpendicular distance from a point \((x_1, y_1)\) to a line \(ax + by + c = 0\) measures how far the point is from the line. We need to find the line with the minimum distance from \((0, 0)\).
Step 2: Key Formula or Approach:
Distance from origin \((0, 0)\) to line \(ax + by + c = 0\) is \(d = \frac{|c|}{\sqrt{a^2 + b^2}}\).
Step 3: Detailed Explanation:
(A) \(3x - 4y + 4 = 0 \implies d_A = \frac{4}{\sqrt{3^2 + (-4)^2}} = \frac{4}{5} = 0.8\).
(B) \(2x - 3y - 5 = 0 \implies d_B = \frac{5}{\sqrt{2^2 + (-3)^2}} = \frac{5}{\sqrt{13}} \approx 1.38\).
(C) \(4x - 3y + 12 = 0 \implies d_C = \frac{12}{\sqrt{4^2 + (-3)^2}} = \frac{12}{5} = 2.4\).
(D) \(5x - 2y - 3 = 0 \implies d_D = \frac{3}{\sqrt{5^2 + (-2)^2}} = \frac{3}{\sqrt{29}} \approx 0.55\).
Comparing distances: \(0.55 \lt 0.8 \lt 1.38 \lt 2.4\).
Thus, line (D) is nearest.
Step 4: Final Answer:
The nearest line is \(5x - 2y = 3\).