Andrew J.

# Calculus question - tangent line equation of ellipse

Question is from Schaum's Guide to Calculus, p.97 q.18:

For the ellipse [MATH]b^2x^2+a^2y^2=a^2b^2[/MATH] show that the equations of its tangent lines of slope m are [MATH]y=mx \pm \sqrt{a^2m^2+b^2}[/MATH]

Question is in chapter on tangent lines and is mostly based on taking implicit derivatives and plugging into point-slope format for the tangent lines. I've seen complicated derivations based on substituting mx+c into the ellipse question and solving for c (by setting the discriminant of the quadratic to zero) but I'm 99.999% certain the book isn't asking for this as that would be far more complex than anything yet covered up to this point.

Really appreciate any help anyone can offer here!

Andrew

Eugene E.

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06/15/20

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Andrew J.

Thank very much Eugene. You are a genius - I wouldn't have figured this out in a million years. Is there a more intuitive way to get there? I had gotten to c=b^2/y_1 but I don't think I every would have known to start working on M^2 from there. Thanks Again!
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06/15/20

Eugene E.

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Sure, no problem! The equation for m contains the factor b^2/y_1, but also contains x_1. The idea is to eliminate x_1 using the equation of the ellipse, which is more readily achieved by squaring both sides of the equation for m. By the way, here is my profile page: https://www.wyzant.com/Tutors/EugeneE
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06/15/20

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