To find the point on the line y = 5x + 4 that is closest to the origin (0, 0), we can use the concept of distance. The distance D from any point (x, y) to the origin can be calculated using the distance formula:
D = √(x² + y²)
However, since our line is defined by the equation y = 5x + 4, we can substitute y in the distance formula. This gives us:
D = √(x² + (5x + 4)²)
To make our calculations easier and to avoid dealing with the square root, we can instead minimize the squared distance:
D² = x² + (5x + 4)²
Expanding (5x + 4)²:
D² = x² + (25x² + 40x + 16) = 26x² + 40x + 16
To find the minimum distance, we take the derivative of D² with respect to x:
d(D²)/dx = 52x + 40
Next, we set the derivative to zero to find the critical points:
52x + 40 = 0
Solving for x gives:
x = -40/52 = -10/13
Now that we have x, we can substitute it back into the line equation to find y:
y = 5(-10/13) + 4 = -50/13 + 52/13 = 2/13
Thus, the point on the line y = 5x + 4 that is closest to the origin is (-10/13, 2/13).
In conclusion, the coordinates of the closest point on the line to the origin are:
Point: (-10/13, 2/13)