To find the point on the line y = 2x + 1 that is closest to the point (5, 2), we can follow several steps:
1. Understanding the Problem
We need to minimize the distance between a point (x, y) on the line and the point (5, 2). The distance D between two points is calculated using the distance formula:
D = √( (x2 – x1)² + (y2 – y1)² )
In our case, this will be:
D = √( (x – 5)² + (y – 2)² )
2. Substitute the Equation of the Line
Since y can be expressed in terms of x using the line equation, substitute y = 2x + 1 into the distance formula:
D = √( (x – 5)² + (2x + 1 – 2)² )
This simplifies to:
D = √( (x – 5)² + (2x – 1)² )
3. Minimize the Distance
To find the minimum distance, we can minimize D² instead of D (since the square root function is monotonically increasing), leading to:
D² = (x – 5)² + (2x – 1)²
Next, expand and simplify this expression:
D² = (x – 5)² + 4x² – 4x + 1
D² = x² – 10x + 25 + 4x² – 4x + 1
D² = 5x² – 14x + 26
4. Calculating the Derivative
Now, we differentiate D² with respect to x and set it to zero to find the critical points:
f'(x) = 10x – 14
Setting the derivative to zero gives:
10x – 14 = 0
10x = 14
x = 1.4
5. Finding the Corresponding y Value
Substitute x = 1.4 back into the equation of the line to find y:
y = 2(1.4) + 1 = 2.8 + 1 = 3.8
6. Conclusion
Thus, the point on the line y = 2x + 1 that is closest to the point (5, 2) is (1.4, 3.8).