Question

Find the point on the line y = 5x + 3 that is closest to the origin. (x, y) =

Answer 1

The point on the line y = 5x + 3 that is closest to the origin is found by determining where a perpendicular line from the origin intersects with it. The point of intersection, which minimizes the distance, is at (-3/26, 63/26).

Step-by-step explanation:

To find the point on the line y = 5x + 3 that is closest to the origin, we can minimize the distance from the origin to any point (x, y) on the line. This distance can be represented by the formula D = √(x2 + y2). Substituting the line equation into the distance formula gives D = √(x2 + (5x + 3)2). To find the minimum distance, we need to differentiate this equation with respect to x and set the derivative equal to zero to solve for x. However, the algebra can be intensive and complex.

To simplify the process, we can use the fact that the shortest distance from a point to a line is along the line perpendicular to it. The slope of the perpendicular line to y = 5x + 3 is -1/5, because perpendicular lines have slopes that are negative reciprocals of each other. The origin has coordinates (0,0), so a line passing through the origin with slope -1/5 is y = -1/5x. To find the intersection of this line with the original line, set the equations equal to each other: 5x + 3 = -1/5x. Solving for x, we get x = -3/26. Plugging this back into either equation gives y = 5(-3/26) + 3 = -15/26 + 78/26 = 63/26. The closest point to the origin on the line y = 5x + 3 is (-3/26, 63/26).

Answer 2

To find the point on the line y = 5x + 3 that is closest to the origin, minimize the distance between the origin and the line. Substitute the equation of the line into the expression for distance squared, simplify, and use the formula x = -b/2a to find the x-coordinate. Then, substitute the x-coordinate back into the equation of the line to find the y-coordinate.

Step-by-step explanation:

To find the point on the line y = 5x + 3 that is closest to the origin, we need to minimize the distance between the origin (0, 0) and the line. The distance between a point (x, y) and the origin is given by the formula d = √(x² + y²). Since we want to minimize d, we can minimize d^2 instead to avoid dealing with square roots. Therefore, we need to minimize the expression d² = x² + y².

We substitute the equation of the line, y = 5x + 3, into the expression d² = x² + y² to get (5x + 3)² + x². Expanding this expression gives 26x² + 30x + 9. To find the minimum of a quadratic function, we can use the equation x = -b/2a, where a = 26, b = 30. Plugging in these values, we find x = -30/52 = -15/26.

Substituting this x value back into the equation of the line, y = 5x + 3, we find y = 5(-15/26) + 3 = -72/26. Therefore, the point on the line y = 5x + 3 that is closest to the origin is approximately (-15/26, -72/26).