Ellipse formulae will help us to solve different types of problems on ellipse in co-ordinate geometry.
1. \(\frac{x^{2}}{a^{2}}\) + \(\frac{y^{2}}{b^{2}}\) = 1, (a > b)
(i) The co-ordinates of the centre are (0, 0).
(ii) The co-ordinates of the vertices are (± a, 0) i.e., (-a, 0) and (a, 0).
(iii) The co-ordinates of the foci are (± ae, 0) i.e., (- ae, 0) and (ae, 0)
(iv) The length of major axis = 2a and the length of minor axis = 2b.
(v) The major axis is along x axis and the equations of major axes is y = 0.
(vi) The minor axis is along y axis and the equations of minor axes is x = 0.
(vii) The equations of the directrices are: x = ± \(\frac{a}{e}\) i.e., x = - \(\frac{a}{e}\) and x = \(\frac{a}{e}\).
(viii) The eccentricity of the ellipse is b\(^{2}\) = a\(^{2}\)(1 - e\(^{2}\)) or, e = \(\sqrt{1 - \frac{b^{2}}{a^{2}}}\).
(ix) The length of the latus rectum 2 ∙ \(\frac{b^{2}}{a}\) = 2a(1 - e\(^{2}\)).
(x) The distance between the two foci = 2ae.
(xi) The distance between two directrices = 2 ∙ \(\frac{a}{e}\).
(xii) Focal distances of a point (x, y) are a ± ex
(xiii) The co-ordinates of the four ends of latera recta are (ae, \(\frac{b^{2}}{a}\)), (ae, -\(\frac{b^{2}}{a}\)), (- ae, \(\frac{b^{2}}{a}\)) and (- ae, -\(\frac{b^{2}}{a}\)).
(xiv) The equations of latera recta are x = ± ae i.e., x = ae and x = -ae.
2. \(\frac{x^{2}}{b^{2}}\) + \(\frac{y^{2}}{a^{2}}\) = 1, (a > b)
(i) The co-ordinates of the centre are (0, 0).
(ii) The co-ordinates of the vertices are (0, ± a) i.e., (0, -a) and (0, a).
(iii) The co-ordinates of the foci are (0, ± ae) i.e., (0, - ae) and (0, ae)
(iv) The length of major axis = 2a and the length of minor axis = 2b.
(v) The major axis is along Y-axis and the equations of major axes is x = 0.
(vi) The minor axis is along X-axis and the equations of minor axes is y = 0.
(vii) The equations of the directrices are: y = ± \(\frac{a}{e}\) i.e., y = - \(\frac{a}{e}\) and y = \(\frac{a}{e}\).
(viii) The eccentricity of the ellipse is b2 = a\(^{2}\)(1 - e\(^{2}\)) or, e = \(\sqrt{1 - \frac{b^{2}}{a^{2}}}\)
(ix) The length of the latus rectum 2 ∙ \(\frac{b^{2}}{a}\) = 2a (1 - e\(^{2}\)).
(x) The distance between the two foci = 2ae.
(xi) The distance between two directrices = 2 ∙ \(\frac{a}{e}\).
(xii) Focal distances of a point (x, y) are a ± ey
(xiii) The co-ordinates of the four ends of latera recta are (\(\frac{b^{2}}{a}\), ae), (-\(\frac{b^{2}}{a}\), ae), (\(\frac{b^{2}}{a}\), -ae) and (-\(\frac{b^{2}}{a}\), -ae).
(xiv) The equations of latera recta are y = ± ae i.e., y = ae and y = -ae.
3. \(\frac{(x - α)^{2}}{a^{2}}\) + \(\frac{(y - β)^{2}}{b^{2}}\) = 1, (a > b)
(i) The co-ordinates of the centre are (α, β).
(ii) The co-ordinates of the vertices are (α ± a, β) i.e., (α - a, β) and (α + a, β).
(iii) The co-ordinates of the foci are (α ± ae, β) i.e., (α - ae, β) and (α + ae, β)
(iv) The length of major axis = 2a and the length of minor axis = 2b.
(v) The major axis is along parallel to x axis and the equations of major axes is y = β.
(vi) The minor axis is along parallel to y axis and the equations of minor axes is x = α.
(vii) The equations of the directrices are: x = α ± \(\frac{a}{e}\) i.e., x = α - \(\frac{a}{e}\) and x = α + \(\frac{a}{e}\).
(viii) The eccentricity of the ellipse is b\(^{2}\) = a\(^{2}\)(1 - e\(^{2}\)) or, e =\(\sqrt{1 - \frac{b^{2}}{a^{2}}}\)
(ix) The length of the latus rectum 2 ∙ \(\frac{b^{2}}{a}\) = 2a (1 - e\(^{2}\)).
(x) The distance between the two foci = 2ae.
(xi) The distance between two directrices = 2 ∙ \(\frac{a}{e}\).
4. \(\frac{(x - α)^{2}}{b^{2}}\) + \(\frac{(y - β)^{2}}{a^{2}}\) = 1, (a > b)
(i) The co-ordinates of the centre are (α, β).
(ii) The co-ordinates of the vertices are (α, β ± a) i.e., (α, β - a) and (α, β + a).
(iii) The co-ordinates of the foci are (α, β ± ae) i.e., (α, β - ae) and (α, β + ae).
(iv) The length of major axis = 2a and the length of minor axis = 2b.
(v) The major axis is along parallel to Y-axis and the equations of major axes is x = α.
(vi) The minor axis is along parallel to X-axis and the equations of minor axes is y = β.
(vii) The equations of the directrices are: y = β ± \(\frac{a}{e}\) i.e., y = β - \(\frac{a}{e}\) and y = β + \(\frac{a}{e}\).
(viii) The eccentricity of the ellipse is b\(^{2}\) = a\(^{2}\)(1 - e\(^{2}\)) or, e = \(\sqrt{1 - \frac{b^{2}}{a^{2}}}\)
(ix) The length of the latus rectum 2 ∙ \(\frac{b^{2}}{a}\) = 2a (1 - e\(^{2}\)).
(x) The distance between the two foci = 2ae.
(xi) The distance between two directrices = 2 ∙ \(\frac{a}{e}\).
5. The point P (x\(_{1}\), y\(_{1}\)) lies outside, on or inside the ellipse \(\frac{x^{2}}{a^{2}}\) + \(\frac{y^{2}}{b^{2}}\) = 1 according as \(\frac{x_{1}^{2}}{a^{2}}\) + \(\frac{y_{1}^{2}}{b^{2}}\) – 1 > 0, = or < 0.
6. If \(\frac{x^{2}}{a^{2}}\) + \(\frac{y^{2}}{b^{2}}\) = 1 is an ellipse, then its auxiliary circle is x\(^{2}\) + y\(^{2}\) = a\(^{2}\).
7. The equations x = a cos ф, y = b sin ф taken together are called the parametric equations of the ellipse \(\frac{x^{2}}{a^{2}}\) + \(\frac{y^{2}}{b^{2}}\) = 1
8. The co-ordinates of the point having eccentric angle ф can be written as (a cos ф, b sin ф). Here (a cos ф, b sin ф) are known as the parametric co-ordinates of the point P.
2nd Grade Math Practice
From Ellipse Formulae to HOME PAGE
Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.
Mar 02, 24 05:31 PM
Mar 02, 24 04:36 PM
Mar 02, 24 03:32 PM
Mar 01, 24 01:42 PM
Feb 29, 24 05:12 PM
New! Comments
Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.