To find the point on the line y = 2x + 3 that is closest to the origin (0, 0), we can use the concept of minimizing the distance from a point to the line.
The distance d between a point (x, y) and the origin (0, 0) can be expressed by the following formula:
d = √((x - 0)² + (y - 0)²) = √(x² + y²)Since we are looking for the point on the line, we can substitute y with 2x + 3 in the distance formula:
d = √(x² + (2x + 3)²)To minimize the distance, we can minimize the square of the distance (to avoid dealing with the square root), which is:
d² = x² + (2x + 3)²Expanding this, we have:
d² = x² + (4x² + 12x + 9) = 5x² + 12x + 9Next, we’ll find the minimum value of this quadratic function. The vertex of a parabola given by the equation ax² + bx + c occurs at:
x = -b / (2a)Here, a = 5 and b = 12, so we have:
x = -12 / (2 * 5) = -12 / 10 = -1.2Now, substituting x = -1.2 back into the line equation to find the corresponding y value:
y = 2(-1.2) + 3 = -2.4 + 3 = 0.6Thus, the point on the line y = 2x + 3 that is closest to the origin (0, 0) is:
(-1.2, 0.6)In conclusion, the closest point on the line to the origin is (-1.2, 0.6).