Conceptualizing two tangent lines to an ellipse

In summary: It sounds like you're looking for an equation that will give you the gradient of the tangent at a particular point on the ellipse. You could try using the formula that Dexter gave you, but it's worth noting that there may be other ways to find the gradient of a tangent, depending on the function.
  • #1
ktpr2
192
0
I'm assuming that given point can only have one tangent line because it's just the instantaneous slope at a point. If so, then how can an ellipse have two tangent lines at a point? Do they mean something else?
 
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  • #2
Does WHO mean something else? Since you haven't told us what the problem says, we can't possibly tell what they might mean! Yes, an ellipse, like any smooth curve has only one tangent line at any point on the ellipse. But it is conceivable that the problem you are thinking of is asking about two tangent lines from some point that is not on the ellipse.
 
  • #3
Perhaps they mean two tangents in opposite directions?
 
  • #4
It can't.A tangent line is a LINE (it's infinite).It comes from [itex]-\infty[/itex] and goes to [itex] +\infty [/itex]...

Daniel.
 
  • #5
I meant vectors...
 
  • #7
Clarification: "Find the equations of both the tangent lines to the ellipse [tex]x^2+4y^2 = 36 [/tex] that pass through the point (12,3)
 
  • #8
That's something else.Plot the ellipse and u'll see that one of the tangents is [tex] y=3 [/tex] and the other can be found from

[tex] \frac{x^{2}}{6^{2}}+\frac{y^{2}}{3^{2}}=1 [/tex] (the ellipse)

[tex] \frac{x-x_{0}}{6^{2}}+\frac{y-y_{0}}{3^{2}}=1 [/tex]
(the tangent to ellipse passing through the point [itex] \left(x_{0},y_{0}\right) [/itex])


Daniel.
 
Last edited:
  • #9
In case it wasn't clear from dexter's response, the answer to your question is that there is only one tangent to an ellipse at a given point on the ellipse, but the point in that problem is well off the ellipse (try plugging it into the equation and you'll see that it doesn't work). To actually answer the question, you can use the formulae dexter gave you.
 
  • #10
ohhhhh So that point is floating off in space and, seeing that, of course there two tangent lines that can run though it on the ellipse. I should've graphed it to better visualize. Thank you both.
 
  • #11
Always plot the graph of the function.The point was away from the ellipse and from it,u could build two tangents...

Daniel.
 
  • #12
Why does it seem so complex? Is it not possible to find the equation of a line from the origin of the ellipse (assumed (0,0)) and then find the perpendicular gradient and then the equation of the tangent or have I missed something that would not make this a feasible method?

The Bob (2004 ©)
 
  • #13
I didn't understand a thing :confused: What gradient?

Daniel.
 
  • #14
I believe my approach is trial and error but from the origin to a point on the ellipse find the gradient and then the perpendicular to that which pass the point (12,3).

I tell you what though, it sounds more stupid the more I think about it.

What do you have to do then?

The Bob (2004 ©)
 
  • #15
Apply the HS taught formula giving the equation of a tangent to an ellipse...?(v.post #8).

Daniel.
 

Related to Conceptualizing two tangent lines to an ellipse

1. What is an ellipse?

An ellipse is a type of geometric shape that looks like a flattened circle. It is defined as the set of all points in a plane, the sum of whose distances from two fixed points (called the foci) is constant.

2. How are two tangent lines conceptualized on an ellipse?

Two tangent lines on an ellipse are lines that touch the ellipse at exactly one point, without intersecting it. These lines are perpendicular to the radius connecting the point of tangency to the center of the ellipse.

3. What is the significance of two tangent lines on an ellipse?

Two tangent lines on an ellipse are important because they help in understanding the properties of the ellipse. They also play a role in determining the equation of an ellipse and its foci.

4. How are the equations of two tangent lines to an ellipse calculated?

The equations of two tangent lines to an ellipse can be calculated using the slope-point form of a line. The slope of the tangent line can be determined by taking the derivative of the ellipse equation at the point of tangency. Then, the point of tangency can be substituted in the slope-point form to obtain the equation of the tangent line.

5. Are two tangent lines always present on an ellipse?

Yes, two tangent lines are always present on an ellipse. This is because the definition of an ellipse involves the foci, which are the two points that the sum of the distances to is constant. Therefore, there will always be two points on the ellipse where the tangent lines are perpendicular to the radius connecting them to the foci.

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