Final answer:
The distance from the origin to the line 4x - 3y = 0 is zero, as it passes through the origin, making it closer than the line 3x - 4y - 4 = 0, which is at a distance of 4/5 from the origin. The correct answer is b. Line 4x - 3y = 0 is closer.
Step-by-step explanation:
To determine which line is closer to the origin, we can find the distance from the origin to the line using the formula D = |Ax0 + By0 + C| / √(A² + B²), where (x0, y0) is the origin (0,0) and Ax + By + C = 0 is the line equation.
For the line 3x - 4y - 4 = 0, the distance from the origin is calculated as D1 = |3(0) - 4(0) - 4| / √(3² + (-4)²) = |(-4)| / √(25) = 4/5.
For the line 4x - 3y = 0, the distance from the origin is calculated as D2 = |4(0) - 3(0) + 0| / √(4² + (-3)²) = 0 / √(25) = 0, which is precisely on the origin.
To check which line is closer to the origin, we can find the distance between the origin and a point on each line. We can choose any convenient point on each line and calculate the distance using the distance formula. Let's choose the point (0,0) to calculate the distance for both lines:
Distance for the line 3x - 4y - 4 = 0: d1 = sqrt((0 - 0)^2 + (-4/(-4))^2) = sqrt(0 + 1) = 1
Distance for the line 4x - 3y = 0: d2 = sqrt((0 - 0)^2 + (0/(-3))^2) = sqrt(0 + 0) = 0
Since d2 is closer to the origin, the correct answer is Line 4x - 3y = 0 is closer.
Therefore, the line 4x - 3y = 0 is closer to the origin since it actually passes through the origin, while the line 3x - 4y - 4 = 0 is at a distance from the origin.
The correct answer is b. Line 4x - 3y = 0 is closer.