Finding the Closest Point on a Line to the Origin
To find the point on the line given by the equation y = 5x – 2 that is closest to the origin (0, 0), we can use the distance formula. The goal is to minimize the distance from any point (x, y) on the line to the origin.
1. Distance Formula
The distance D from the origin to a point (x, y) is given by:
D = √(x2 + y2)
2. Substitute the Line Equation
Since we know that y can be expressed in terms of x from the line equation, we substitute y = 5x – 2 into the distance formula:
D = √(x2 + (5x – 2)2)
3. Simplifying the Distance
Now, let’s simplify the distance expression:
D = √(x2 + (25x2 – 20x + 4))
D = √(26x2 – 20x + 4)
4. Minimize D²
To find the minimum distance, it’s easier to minimize D² (the expression inside the square root) instead:
D² = 26x2 – 20x + 4
5. Finding the Derivative
Taking the derivative with respect to x:
f'(x) = 52x – 20
6. Setting the Derivative to Zero
Setting the derivative equal to zero:
52x – 20 = 0
52x = 20
x = 20/52 = 5/13
7. Finding y
Now we substitute x = 5/13 back to the line equation to find y:
y = 5(5/13) – 2 = 25/13 – 2 = 25/13 – 26/13 = -1/13
8. Closest Point Coordinates
The coordinates of the point on the line that is closest to the origin are:
(5/13, -1/13)
Conclusion
In summary, the point on the line y = 5x – 2 closest to the origin is at (5/13, -1/13).