# When will the line intercept the ellipse a second time?

## 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?

???

## 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.

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D H
Staff Emeritus
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.

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.

HallsofIvy