To find the point on the line y = 4x + 3 that is closest to the origin (0, 0), we can use the concept of minimizing the distance between the point on the line and the origin.
The distance D from the origin to a point (x, y) on the line can be given by the formula:
D = √(x² + y²)
Since y is defined by the line equation, we can substitute it in:
D = √(x² + (4x + 3)²)
To simplify the calculation, we can minimize the square of the distance (D²) instead:
D² = x² + (4x + 3)²
Expanding this gives:
D² = x² + (16x² + 24x + 9) = 17x² + 24x + 9
Now, we want to minimize D². To find the minimum point, we can take the derivative and set it to zero:
d(D²)/dx = 34x + 24 = 0
Solving for x, we find:
34x + 24 = 0
x = -24/34 = -12/17
Next, we can substitute this value of x back into the line equation to find y:
y = 4(-12/17) + 3 = -48/17 + 51/17 = 3/17
Thus, the point on the line y = 4x + 3 that is closest to the origin is:
(-12/17, 3/17)
This gives us the coordinates of the closest point on the specified line to the origin.