The tangent and the normal to the conic

In summary, the tangent and normal to the conic at a point (a\cos\left({\theta}\right), b\sin\left({\theta}\right)) meet at P and P' on the major axis, where PP'=a. We need to show that e^2cos^2\theta + cos\theta -1 = 0, where e is the eccentricity of the conic.
  • #1
debrajr
4
0
The tangent and the normal to the conic
\(\displaystyle \frac{x^2}{a^2}+\frac{y^2}{b^2}=1\)
at a point \(\displaystyle (a\cos\left({\theta}\right), b\sin\left({\theta}\right))\)
meet the major axis in the points \(\displaystyle P\) and \(\displaystyle P'\), where \(\displaystyle PP'=a\)
Show that \(\displaystyle e^2cos^2\theta + cos\theta -1 = 0\), where \(\displaystyle e\) is the eccentricity of the conic
 
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  • #2
Hello debrajr and welcome to MHB! :D

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Related to The tangent and the normal to the conic

1. What is the difference between the tangent and the normal to a conic?

The tangent to a conic at a given point is a line that touches the curve at that point and has the same slope as the curve at that point. The normal, on the other hand, is a line that is perpendicular to the tangent at that point. In other words, the tangent and the normal are two different lines that are related to each other through their slopes and angles.

2. How do you find the equation of the tangent and the normal to a conic?

To find the equation of the tangent at a given point on a conic, you can use the derivative of the equation of the conic to calculate the slope of the tangent at that point. Then, you can use the point-slope form of a line to write the equation of the tangent. Similarly, to find the equation of the normal, you can use the negative reciprocal of the slope of the tangent as the slope of the normal and follow the same steps as finding the equation of the tangent.

3. Can a conic have multiple tangents and normals at a single point?

No, a conic can only have one tangent and one normal at a given point. This is because the slope of a curve is a unique value at any point, and therefore, there can only be one line that has the same slope as the curve at that point (tangent) and one line that is perpendicular to it (normal).

4. Do all conics have tangents and normals?

Yes, all conics have tangents and normals at every point on the curve. This is because the equations of conics are continuous and differentiable, meaning that the slope of the curve exists at every point and can be used to find the tangent and normal lines.

5. How are the tangent and normal related to the curvature of a conic?

The tangent and normal lines can provide information about the curvature of a conic at a given point. The curvature of a curve is a measure of how much the curve is bending at a point, and it can be calculated using the slope of the tangent and the second derivative of the curve. The tangent and normal are also perpendicular to each other, and their intersection at a point represents the center of curvature of the curve at that point.

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