Find the eccentricity of this ellipse

In summary, the equation for the tangent at point P is xx_2 + yy_2 - c^2 = 0. The equation for the tangent at point A is xx_1 + yy_1 - c^2 = 0. When these two equations are solved, x = \dfrac{c^2 (y_2 - y_1)}{x_1y_2-x_2y_1} and y = \dfrac{c^2 (x_1 - x_2)}{x_1y_2-x_2y_1}.
  • #1
utkarshakash
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Homework Statement


The tangent at any point P of a circle meets the tangent at a fixed point A in T, and T is joined to B, the other end of diameter through A. Prove that the locus of point of intersection of AP and BT is an ellipse whose eccentricity is [itex] 1/ \sqrt{2}[/itex]

Homework Equations



The Attempt at a Solution


The very first thing I do is assume the equation of a circle. The next thing is to write the equations for tangents and solve them to get T. But it is getting complicated as nothing is known to me. So there are a number of variables which can't be eliminated. Any other ideas?
 
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  • #2
hi utkarshakash! :smile:
utkarshakash said:
The very first thing I do is assume the equation of a circle. The next thing is to write the equations for tangents and solve them to get T. But it is getting complicated as nothing is known to me. So there are a number of variables which can't be eliminated.

show us what you've tried, and where you're stuck, and then we'll know how to help! :smile:

alternatively, since the eccentricity is 1/√2, the minor axis must be 1/√2 time the major axis …

so have you tried squashing the whole diagram by 1/√2 (along AB), so that the the final result is a circle? :wink:
 
  • #3
tiny-tim said:
hi utkarshakash! :smile:


show us what you've tried, and where you're stuck, and then we'll know how to help! :smile:

alternatively, since the eccentricity is 1/√2, the minor axis must be 1/√2 time the major axis …

so have you tried squashing the whole diagram by 1/√2 (along AB), so that the the final result is a circle? :wink:

For the sake of simplicity, let the equation of the circle be [itex]x^2 + y^2 = c^2 [/itex].

Let A = (x1,y1) and P = (x2,y2)

Equation of tangent at P
[itex] xx_2 + yy_2 - c^2 = 0[/itex]
Equation of tangent at A
[itex] xx_1 + yy_1 -c^2 = 0[/itex]

When I solve these two equations I get

[itex] x = \dfrac{c^2 (y_2 - y_1)}{x_1y_2-x_2y_1} \\

y = \dfrac{c^2 (x_1 - x_2)}{x_1y_2-x_2y_1}[/itex]

OMG It looks so dangerous! I don't want to proceed ahead as the calculation will be complex and rigorous.
 
  • #4
utkarshakash said:
Let A = (x1,y1) and P = (x2,y2)

no!

A is fixed, so you can simplify by letting A = (c,0) :wink:

(and P = (ccosθ,csinθ) )
 
Last edited:
  • #5
tiny-tim said:
no!

A is fixed, so you can simplify by letting A = (c,0) :wink:

(and P = (ccosθ,csinθ) )

That did simplify the expression to a large extent. Thanks!
 

FAQ: Find the eccentricity of this ellipse

1. What is an ellipse?

An ellipse is a geometric shape that resembles a flattened circle. It is defined as the set of all points in a plane whose distances from two fixed points, called foci, are constant.

2. How do you find the eccentricity of an ellipse?

The eccentricity of an ellipse is found by dividing the distance between the two foci by the length of the major axis (the longest diameter of the ellipse). The resulting value ranges from 0 to 1, with 0 representing a circle and 1 representing a line.

3. What is the significance of the eccentricity of an ellipse?

The eccentricity of an ellipse is a measure of how "stretched out" the ellipse is. It can determine the shape of the ellipse and its relationship to other conic sections. Ellipses with higher eccentricity values tend to be more elongated, while those with lower values are closer to a circle.

4. Can the eccentricity of an ellipse be greater than 1?

No, the eccentricity of an ellipse cannot be greater than 1. This would mean that the distance between the two foci is greater than the length of the major axis, which is not possible in an ellipse.

5. How does the eccentricity of an ellipse affect its orbit around a focus?

The eccentricity of an ellipse determines the shape and orientation of its orbit around a focus. Ellipses with lower eccentricity values have more circular orbits, while those with higher values have more elongated and elliptical orbits.

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