To find the point on the line y = 5x + 3 that is closest to the origin (0, 0), we first need to understand the formula for the distance between a point and the origin. The distance D from any point (x, y) to the origin can be expressed as:
D = √(x² + y²)
However, to simplify our calculations, we can minimize the square of the distance instead:
D² = x² + y²
Substituting y = 5x + 3 into the distance formula gives:
D² = x² + (5x + 3)²
Expanding this, we have:
D² = x² + (25x² + 30x + 9)
This simplifies to:
D² = 26x² + 30x + 9
Next, to find the minimum of this quadratic function, we can use the vertex formula x = -b/(2a). Here, a = 26 and b = 30, so:
x = -30 / (2 * 26) = -15 / 26
Now, we can substitute this value of x back into the line equation to find the corresponding y value:
y = 5(-15/26) + 3 = -75/26 + 78/26 = 3/26
Thus, the point on the line y = 5x + 3 that is closest to the origin is:
(-15/26, 3/26)