Question

Find an equation of the normal line to the parabola y=x² −8x+2 that is paraliel to the line x−4y=4. y=

Answer 1

The equation of the normal line to the parabola y = x² − 8x + 2 that is parallel to the line x − 4y = 4 is y = 2x + 6.

Step-by-step explanation:

To find the equation of the normal line to the parabola that's parallel to the given line, we need to identify the slope of the line x - 4y = 4. This equation can be rewritten in slope-intercept form
`(y = mx + c)` to identify its slope. Rearranging the equation, we get y = 0.25x - 1. Then, the slope of this line is 0.25.

The slope of the normal line to a curve at a given point is the negative reciprocal of the slope of the tangent at that point. For the parabolic curve y = x² − 8x + 2, we need to find where on the curve the tangent has a slope of 0.25. The derivative of y with respect to x gives the slope of the tangent at any point on the curve. Taking the derivative of y = x² − 8x + 2, we get y' = 2x - 8.

Setting y' = 0.25 and solving for x gives x = 4.25. Substituting this x-value back into the original equation y = x² − 8x + 2, we find y = 4.25² - 8 * 4.25 + 2 = 18.0625 - 34 + 2 = -13.9375. Therefore, the point of tangency is (4.25, -13.9375).

The slope of the normal line is the negative reciprocal of the slope of the tangent, which is -1/0.25 = -4. Hence, using the point-slope form with the point of tangency (4.25, -13.9375), the equation of the normal line becomes y - (-13.9375) = -4(x - 4.25). Simplifying this equation results in y = -4x + 18 + 13.9375, which simplifies to y = -4x + 31.9375. Hence, the final equation of the normal line parallel to x - 4y = 4 is y = 2x + 6.

Answer 2

To determine the equation of the normal line to the parabola y=x² - 8x + 2, which is parallel to the line x - 4y = 4, first find the slope of the given line (¼). Then, find where the slope of the parabola's tangent is -4, leading to the point (2, -10). Finally, use the slope (¼) and point (2, -10) to write the normal line's equation: y = ¼x - 10.5.

Step-by-step explanation:

To find an equation of the normal line to the parabola y=x² − 8x + 2 that is parallel to the line x − 4y = 4, we first need to find the slope of the given line.

We can rewrite the line's equation in slope-intercept form as y = ¼x - 1.

This reveals that the slope of the line is ¼.

Because the normal line we want to find is parallel to this line, it will also have a slope of ¼.

Next, we take the derivative of the parabola's equation to find the slope at any point on the parabola, which gives y' = 2x - 8.

We need to find the value of x where the slope of the tangent to the parabola is the negative reciprocal of ¼, which is -4, since the normal line is perpendicular to the tangent.

By setting 2x - 8 = -4, we solve for x and find that x = 2.

After finding the x-coordinate, we substitute x = 2 into the original parabola equation to get the y-coordinate, which gives us y = 2² - 8(2) + 2 = -10.

With the point (2, -10) on the normal line and the slope of ¼, we use the point-slope form of a line to write the equation of the normal line. The equation is y + 10 = ¼(x - 2).

Simplifying, we get the final equation of the normal line: y = ¼x - ¼(2) - 10,

which simplifies to y = ¼x - ¼ - 10, or y = ¼x - 10.5.