Question

Find the point on the line y = 4x 5 that is closest to the origin. (x, y) =

Answer 1

To find the point on the line y = 4x + 5 that is closest to the origin, we can use the distance formula.

Step-by-step explanation:

To find the point on the line y = 4x + 5 that is closest to the origin, we can use the distance formula. The distance between the origin (0,0) and a point (x, y) is given by the formula d = sqrt((x - 0)^2 + (y - 0)^2). Since we are looking for the point on the line, we can substitute y = 4x + 5 into the distance formula to get d = sqrt(x^2 + (4x + 5)^2). To minimize the distance, we can find the minimum value of d by taking the derivative and setting it equal to 0. Solving for x gives the x-coordinate of the point on the line closest to the origin. Substituting this value of x into y = 4x + 5 will give the y-coordinate.

Answer 2

To find the point on the line y = 4x + 5 that is closest to the origin, find the perpendicular line from the origin to the given line, set the equations equal to each other, solve for x, and then solve for y. The point closest to the origin is at (-20/17, 5/17).

Step-by-step explanation:

The given line has the equation y = 4x + 5, and we want to find the point on this line that is closest to the origin. To solve this, we make use of the fact that the shortest distance from a point to a line is along the perpendicular to the line. Therefore, we need to find a line perpendicular to y = 4x + 5 that passes through the origin (0,0). The slope of any line perpendicular to this one must be the negative reciprocal of 4, which is -1/4. So, the equation of the perpendicular line is y = -1/4x. To find the intersection of these two lines, we set them equal to each other and solve for x:

4x + 5 = -1/4x

Multiplying through by 4 to eliminate the fraction:
16x + 20 = -x

Combine like terms:
17x + 20 = 0

Solving for x gives us x = -20/17. We then solve for y using either line's equation:
y = 4(-20/17) + 5

Which simplifies to y = -80/17 + 85/17, resulting in y = 5/17. So, the point on the line closest to the origin is (-20/17, 5/17).