To find the point on the line y = 5x + 2 that is closest to the origin (0, 0), we can use the concept of distance from a point to a line.
The distance D from a point (x, y) to the origin is given by:
D = √(x² + y²)
However, since we want to minimize the distance, we can minimize the square of the distance to avoid dealing with the square root:
D² = x² + y²
Substituting the equation of the line into our distance formula, we have:
D² = x² + (5x + 2)²
Now, expand the equation:
D² = x² + (25x² + 20x + 4)
D² = 26x² + 20x + 4
To find the minimum distance, we can take the derivative of D² with respect to x and set it to zero:
d(D²)/dx = 52x + 20 = 0
Solve for x:
52x = -20
x = -20/52 = -5/13
Now, plug x back into the line equation to find y:
y = 5(-5/13) + 2
y = -25/13 + 2 = -25/13 + 26/13 = 1/13
Thus, the point on the line y = 5x + 2 that is closest to the origin is:
(-5/13, 1/13)