Question

Find an equation of the normal line to the curve of y = sqrt(x) that is parallel to the line 4x y = 1.

Answer 1

To find the equation of the normal line to the curve y = sqrt(x) that is parallel to the line 4xy = 1, we need to find the slope of the normal line and a point on the curve. The slope of the normal line is the negative reciprocal of the slope of the given line, and a point on the curve can be found by substituting an x-value into the curve equation. Using the point-slope form of a linear equation, we can write the equation of the normal line. Simplifying the equation gives us the equation of the normal line to the curve y = sqrt(x) that is parallel to the line 4xy = 1 as y = -4x² + 4x + 1.

Step-by-step explanation:

To find the equation of the normal line to the curve of y = sqrt(x) that is parallel to the line 4x y = 1, we need to find the slope of the normal line and a point on the curve. The slope of the normal line is the negative reciprocal of the slope of the given line. The slope of the given line is 1/4x, so the slope of the normal line is -4x. To find a point on the curve, we can choose any x-value and substitute it into the curve equation to find the corresponding y-value. Let's choose x = 1. Substituting this into the curve equation, we get y = sqrt(1) = 1. Therefore, the point on the curve is (1, 1).

Using the point-slope form of a linear equation, we can write the equation of the normal line as y - y1 = m(x - x1), where (x1, y1) is the point on the curve and m is the slope of the normal line. Substituting the values we have, we get y - 1 = -4x(x - 1). Simplifying this equation gives us the equation of the normal line to the curve of y = sqrt(x) that is parallel to the line 4x y = 1 as y = -4x² + 4x + 1.

Answer 2

The equation of the normal line to the curve y = √x that is parallel to the line 4xy = 1, follow these steps: find the derivative of y = √x, use the fact that the normal line is perpendicular to the tangent line, the equation of the normal line to the curve y = √x that is parallel to the line 4xy = 1 is y = -x + 3/4.

The equation of the normal line to the curve y = √x that is parallel to the line 4xy = 1, we'll follow these steps:

Find the derivative of y = √x to get the slope of the tangent line.

Use the fact that the normal line is perpendicular to the tangent line, so the slope of the normal line is the negative reciprocal of the slope of the tangent line.

Set up an equation for the line with the calculated slope that is parallel to 4xy = 1.

Solve for the intersection point of this line and y = √x.

Therefore, the equation of the normal line to the curve y = √x that is parallel to the line 4xy = 1 is y = -x + 3/4.