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Thread: Clarification on Ellipse (Implicit Differentiation)

  1. #1
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    Clarification on Ellipse (Implicit Differentiation)

    Clarification on Ellipse (Implicit Differentiation)-capture.png
    Hey guys,

    so i know how to find the points of where the horizontal and vertical asymptotes for the ellipse.

    So, instead of going through all the work, i just want to clarify something.

    Let's say the point of the horizontal asymptote is like (-3, 1) and (4,5) and the vertical asymptote is like (-3, 2) and (6,3) .

    Even if it doesn't make sense, do i get the distance by using the distance formula for each of the pair of points, so i can ultimately calculate the dimensions?
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  2. #2
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    re: Clarification on Ellipse (Implicit Differentiation)

    No. If you notice the box does not have the same dimensions as the semi-minor and semi-major axes.

    Suppose the two points that are tangent to the top and bottom of the box are $(x_t, y_t),~(x_b, y_b)$

    Then the height of the box is $y_t - y_b$

    The $x$ values here tell you nothing.

    The width of the box will be the difference between the $x$ values of the tangent points at the sides.
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  3. #3
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    re: Clarification on Ellipse (Implicit Differentiation)

    First, "asymptotes" is the wrong word here. An ellipse does NOT HAVE "asymptotes" and the problem you post does not use that word. The problem asks for "tangents" to the ellipse.

    The ellipse is given by $x^2- xy+ y^2= 3$. Differentiating, $2x- xy'- y+ 2yy'= 0$. $(2y- x)y'= y- 2x$ so $y'= \frac{y- 2x}{2y- x}$. A tangent line will be horizontal to the ellipse if and only if $\frac{y- 2x}{2y- x}= 0$ and vertical if that fraction does not exist. The first is true if and only if y- 2x= 0 and the second if and only if 2y- x= 0. So the horizontal lines tangent to the ellipse satisfy y- 2x= 0 and $x^2- xy+ y^2= 3$. Vertical lines tangent to the ellipse satisfy 2y- x= 0 and $x^2- xy+ y^2= 3$.
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  4. #4
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    re: Clarification on Ellipse (Implicit Differentiation)

    Thank you. That makes more sense the way you put it! How would i describe the location of the box? I can't use the points of the horizontal and vertical asymptotes can I? It wouldn't be the corner points? So, do i just find the intersection of the horizontal and vertical tangent lines?
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  5. #5
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    re: Clarification on Ellipse (Implicit Differentiation)

    Again, those are not "asymptotes", they are tangents to the ellipse. The horizontal lines are of the form y= constant, the vertical lines are of the form x= constant.
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