When will the line intercept the ellipse a second time?

In summary, the equation of the perpendicular line to the ellipse x^2 − xy + y^2 = 3 at the point (-1,1) is y = -x and it intersects the ellipse again at (1,-1). This can be proven by replacing the y values in the ellipse equation with -x and solving for x. The other solution for x is 1, resulting in the point (1,-1) on the ellipse.
  • #1
diffusion
73
0

Homework Statement


Find the equation of the line perpendicular to the ellipse x^2 − xy + y^2 = 3 at the point (-1,1). Where does the perpendicular line intercept the ellipse a second time?

Homework Equations


?

The Attempt at a Solution


I have already found the equation of the perpendicular line (f(x) = x - 2), just not sure how prove mathematically where it intercepts again. Of course I can just look at the graph, but I should be able to do it without referring to the graph at all. The slope of the line BTW is -1.
 
Physics news on Phys.org
  • #2
Have you tried drawing a picture?

Does f(x)=x-2 have a slope of -1? (In other words, your perpendicular is not correct.)

Instead of f(x), use y. Now you have two equations in two unknowns.
 
  • #3
D H said:
Have you tried drawing a picture?

Does f(x)=x-2 have a slope of -1? (In other words, your perpendicular is not correct.)

Instead of f(x), use y. Now you have two equations in two unknowns.

Sorry, I just realized I made a mistake. The equation for the line PARALLEL to (-1,1) is f(x)=x-2. Sorry.

The equation for the perpendicular line is simply f(x) = -x

That said, since the perpendicular line passes through (-1,1) and then through the origin, you know that it intercepts the ellipse again at (1,-1). So I know the answer, just not sure how to justify it that mathematically.

My best justification would be, "Since every point on an ellipse has a corresponding point in which the x and y values are reversed, then we know that the perpendicular line passes again through the ellipse at (1, -1)."

Something like that, but again it would be better if I could prove this via calculations and not sentences.
 
  • #4
You know the perpendicular line is y= -x and you want to determine where it crosses x2[/sub]- xy+ y2= 3. Okay, replace each y with -x to get x2- x(-x)+ (-x)2= 3x2= 3 and solve for x. You already know one solution is x= -1 since (-1, 1) is on the ellipse. What is the other solution?
 

1. What is the mathematical equation for a line intercepting an ellipse a second time?

The mathematical equation for a line intercepting an ellipse a second time is given by:
ax + by + c = 0
Where, a and b are the coefficients of x and y respectively, and c is a constant term.

2. How can I determine the coordinates of the second intersection point between a line and an ellipse?

To determine the coordinates of the second intersection point, you can use the substitution method. First, substitute the equation of the line into the equation of the ellipse. This will give you a quadratic equation in terms of x. Solve for the value(s) of x and then use those values to find the corresponding y coordinates.

3. Is it possible for a line to not intersect an ellipse a second time?

Yes, it is possible for a line to not intersect an ellipse a second time. This can happen if the line is tangent to the ellipse or if the line is parallel to the ellipse and does not intersect it at all.

4. Can the coordinates of the second intersection point be imaginary?

Yes, the coordinates of the second intersection point can be imaginary. This can happen if the line does not intersect the ellipse in the real plane, but intersects it in the complex plane. In this case, the coordinates of the intersection point will be complex numbers.

5. How does the position and slope of the line affect the second intersection point with an ellipse?

The position and slope of the line can affect the second intersection point with an ellipse in different ways. For example, if the line is parallel to the major axis of the ellipse, it will intersect the ellipse at two real points. However, if the line is parallel to the minor axis, it will only intersect the ellipse at one real point. The slope of the line also affects the angle at which it intersects the ellipse, which in turn can affect the coordinates of the second intersection point.

Similar threads

  • Precalculus Mathematics Homework Help
Replies
6
Views
405
  • Precalculus Mathematics Homework Help
Replies
11
Views
2K
  • Precalculus Mathematics Homework Help
Replies
17
Views
986
  • Precalculus Mathematics Homework Help
Replies
10
Views
2K
  • Precalculus Mathematics Homework Help
Replies
10
Views
604
  • Precalculus Mathematics Homework Help
Replies
9
Views
2K
  • Precalculus Mathematics Homework Help
Replies
10
Views
2K
  • Precalculus Mathematics Homework Help
Replies
8
Views
2K
  • Set Theory, Logic, Probability, Statistics
Replies
4
Views
880
  • Precalculus Mathematics Homework Help
Replies
2
Views
2K
Back
Top