To find the point on the line y = 2x + 3 that is closest to the origin (0, 0), we can use the concept of distance. The distance d from a point (x, y) to the origin is given by the formula:
d = √(x² + y²)
Substituting the equation of the line into this formula, we have:
d = √(x² + (2x + 3)²)
Now, squaring both sides to simplify our calculation (we can minimize the square of the distance instead), we get:
d² = x² + (2x + 3)²
Expanding this gives us:
d² = x² + (4x² + 12x + 9) = 5x² + 12x + 9
To find the minimum distance, we need to minimize d² which is a quadratic function. We’ll take the derivative of d² with respect to x and set it to zero:
f'(x) = 10x + 12
Setting this equal to zero gives:
10x + 12 = 0
Solving for x, we find:
x = -1.2
Next, we need to substitute this value back into the line equation to find the corresponding y value:
y = 2(-1.2) + 3 = -2.4 + 3 = 0.6
Thus, the point on the line y = 2x + 3 that is closest to the origin is:
(-1.2, 0.6)