To find the point on the line y = 4x + 5 that is closest to the origin (0, 0), we can use the method of minimizing the distance from the origin to any point on the line.
The distance d from the origin to a point (x, y) on the line can be expressed as:
d = √(x² + y²)
However, since d is a square root, we can minimize d² instead for simplicity:
d² = x² + y²
Substituting the line equation y = 4x + 5 into this distance formula gives us:
d² = x² + (4x + 5)²
Expanding this, we have:
d² = x² + (16x² + 40x + 25)
d² = 17x² + 40x + 25
Next, we find the minimum value of this quadratic function. The x-coordinate of the vertex of a parabola described by the equation ax² + bx + c is given by -b/(2a). Here, a = 17 and b = 40, so:
x = -40 / (2 * 17) = -40 / 34 = -20 / 17
Now, we can substitute this x-value back into the line equation to find the corresponding y-coordinate:
y = 4(-20/17) + 5 = -80/17 + 85/17 = 5/17
Thus, the point on the line y = 4x + 5 that is closest to the origin is:
(-20/17, 5/17)