Parabola, tangebt line and Normal Line intersect

In summary, the normal line to the parabola y= x - x^2 at the point (1,0) intersects the parabola again at the point (2,-1), as the equations y= x - x^2 and y = x - 1 are set equal to find the point of intersection.
  • #1
mrm0607
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0

Homework Statement



Where does the normal line to the parabola y= x - x^2 at the point (1,0) intersect the parabola a second time? Illustrate with a sketch

Homework Equations



y= x - x^2

The Attempt at a Solution



y' = 1-2x

slope of tangent = -1 (after substituting value of x in above eq)

-ve reciprocal of (-1) = 1

equation of normal line is

y - 0 = 1 (x -1)

y = x -1

After this I am not sure how to find the second point of intersection.

I understand there has to be one since parabola is symmetric about its axis.

Thank you.
 
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  • #2
You've done everything correct so far. You now have a separate equation and you want to find where it intersects the parabola. By setting the equations equal, you will get your point of intersection.
 
  • #3
Thank you
 

Related to Parabola, tangebt line and Normal Line intersect

1. What is the definition of a parabola?

A parabola is a symmetrical curve formed by the intersection of a cone and a plane parallel to its side. It is a type of quadratic function that has the form y = ax^2 + bx + c, where a, b, and c are constants and a ≠ 0.

2. How do you find the tangent line to a parabola at a specific point?

To find the tangent line to a parabola at a specific point, you need to take the derivative of the parabola equation and then substitute the x-coordinate of the point into the derivative. This will give you the slope of the tangent line, which can then be used to find the equation of the line using the point-slope form.

3. What is the normal line to a parabola?

The normal line to a parabola is a line that is perpendicular to the tangent line at a specific point on the parabola. It intersects the parabola at that point and has a slope that is the negative reciprocal of the tangent line's slope.

4. Can a parabola and its tangent line intersect at more than one point?

No, a parabola and its tangent line can only intersect at one point. This is because the tangent line is defined as a line that touches the parabola at only one point and has the same slope as the parabola at that point. Any other point of intersection would result in a different slope, making it not a tangent line.

5. How do you determine if a point lies on the normal line to a parabola?

To determine if a point lies on the normal line to a parabola, you can use the slope formula to find the slope between the point and the point of intersection between the tangent line and the parabola. If this slope is the negative reciprocal of the tangent line's slope, then the point lies on the normal line.

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