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

Nearest Line to the Origin

A line is nearest to the origin if it has the smallest perpendicular distance from the origin, calculated by $d = |C| / \sqrt{A^2 + B^2}$ for the standard form $Ax + By + C = 0$.

For $3x - 4y + 4 = 0$: $d = |4| / \sqrt{9 + 16} = 4/5 = 0.8$

For $2x - 3y - 5 = 0$: $d = | -5 | / \sqrt{4 + 9} \approx 5 / 3.605 \approx 1.387$

For $4x - 3y + 12 = 0$: $d = |12| / \sqrt{16 + 9} = 12/5 = 2.4$

For $5x - 2y - 3 = 0$: $d = | -3 | / \sqrt{25 + 4} \approx 3 / 5.385 \approx 0.557$

Distances: 0.8, 1.387, 2.4, 0.557. The least is approximately $0.557$.

The line $5x - 2y = 3$ is nearest to the origin.