To find the point on the line y = 3x + 4 that is closest to the origin (0, 0), we will use the concept of minimizing the distance from the origin to any point on the line.
The distance d from the origin to a point (x, y) is given by the formula:
d = √(x² + y²)
However, instead of minimizing d, we will minimize d² to simplify calculations. Thus, we express d² as:
d² = x² + y²
Since any point on the line can be described in terms of x, we can substitute y from the line equation:
d² = x² + (3x + 4)²
Now expanding this:
d² = x² + (9x² + 24x + 16)
d² = 10x² + 24x + 16
Next, we take the derivative of d² with respect to x and set it to zero to find the minimum:
d²’ = 20x + 24
Setting this equal to zero:
20x + 24 = 0
20x = -24
x = -24 / 20 = -1.2
We can now find the corresponding y value by substituting x back into the line equation:
y = 3(-1.2) + 4
y = -3.6 + 4
y = 0.4
Thus, the point on the line y = 3x + 4 that is closest to the origin is (-1.2, 0.4).