Vertical Tangent Lines to an Ellipse

In summary, a vertical tangent line to an ellipse is a line that is perpendicular to the horizontal axis of the ellipse at a specific point on the ellipse. The equation for this line can be determined by taking the derivative of the ellipse's equation and setting it equal to infinity. An ellipse can have multiple vertical tangent lines, and their occurrence is related to the eccentricity of the ellipse. In real-world applications, vertical tangent lines to an ellipse can represent maximum or minimum values and have uses in fields such as orbital mechanics and engineering.
  • #1
unf0r5ak3n
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Homework Statement



Consider the curve x^2+xy+y^2=27

Homework Equations


Find all points on the curve where the lines tangent to the curve are vertical


The Attempt at a Solution


I found dy/dy = (-2x-y)/x+2y)

and I think I found the equations of lines visually to be x=6 and x=-6 but I'm not sure how to show the work.
 
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  • #2
So you'll need to find what points on the curve satisfy x+2y=0...
 
  • #3
oh wow i didnt realize that thanks
 

1. What is a vertical tangent line to an ellipse?

A vertical tangent line to an ellipse is a line that is perpendicular to the horizontal axis of the ellipse at a specific point on the ellipse. This means that the slope of the tangent line at that point is undefined, and the line is vertical.

2. How is the equation for a vertical tangent line to an ellipse determined?

The equation for a vertical tangent line to an ellipse can be determined by taking the derivative of the equation of the ellipse and setting it equal to infinity. This will give the x-coordinate of the point of tangency, and the equation can be written as x = a or x = -a, depending on the direction of the ellipse.

3. Can an ellipse have more than one vertical tangent line?

Yes, an ellipse can have multiple vertical tangent lines. This will occur when the ellipse is elongated or stretched in the horizontal direction, causing the tangent lines to be closer together and intersect the ellipse at multiple points.

4. How are vertical tangent lines to an ellipse related to its eccentricity?

The eccentricity of an ellipse is a measure of how elongated or circular it is. When the eccentricity is close to 0, the ellipse is nearly circular and will not have any vertical tangent lines. As the eccentricity increases, the ellipse becomes more elongated and will have multiple vertical tangent lines.

5. What is the significance of vertical tangent lines to an ellipse in real-world applications?

In real-world applications, vertical tangent lines to an ellipse can represent maximum or minimum values in a variety of situations. For example, in orbital mechanics, the paths of planets around the sun can be described by ellipses, and the points where the tangent line is vertical can represent the closest or farthest distance from the sun. In engineering, the shape of ellipses can be used in the design of lenses and mirrors for optical instruments.

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