Ellipse Formulae

Ellipse formulae will help us to solve different types of problems on ellipse in co-ordinate geometry.

1. x2a2 + y2b2 = 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 = ± ae i.e., x = - ae and x = ae.

(viii) The eccentricity of the ellipse is b2 = a2(1 - e2) or, e = 1b2a2.

(ix) The length of the latus rectum 2 b2a = 2a(1 - e2).

(x) The distance between the two foci = 2ae.

(xi) The distance between two directrices = 2 ae.

(xii) Focal distances of a point (x, y) are a ± ex

(xiii) The co-ordinates of the four ends of latera recta are (ae, b2a), (ae, -b2a), (- ae, b2a) and (- ae, -b2a).

(xiv) The equations of latera recta are x = ± ae i.e., x = ae and x = -ae.

                      

2. x2b2 + y2a2 = 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 = ± ae i.e., y = - ae and y = ae.

(viii) The eccentricity of the ellipse is b2 = a2(1 - e2) or,  e = 1b2a2

(ix) The length of the latus rectum 2 b2a = 2a (1 - e2).

(x) The distance between the two foci = 2ae.

(xi) The distance between two directrices = 2 ae.

(xii) Focal distances of a point (x, y) are a ± ey

(xiii) The co-ordinates of the four ends of latera recta are (b2a, ae), (-b2a, ae), (b2a, -ae) and (-b2a, -ae).

(xiv) The equations of latera recta are y = ± ae i.e., y = ae and y = -ae.

3. (xα)2a2 + (yβ)2b2 = 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 ± ae i.e., x = α - ae and x = α + ae.

(viii) The eccentricity of the ellipse is b2 = a2(1 - e2) or, e =1b2a2

(ix) The length of the latus rectum 2  b2a = 2a (1 - e2).

(x) The distance between the two foci = 2ae.

(xi) The distance between two directrices = 2  ae.


4. (xα)2b2 + (yβ)2a2 = 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 ± ae i.e., y = β - ae and y = β + ae.

(viii) The eccentricity of the ellipse is b2 = a2(1 - e2) or, e = 1b2a2

(ix) The length of the latus rectum 2  b2a = 2a (1 - e2).

(x) The distance between the two foci = 2ae.

(xi) The distance between two directrices = 2  ae.


5. The point P (x1, y1) lies outside, on or inside the ellipse x2a2 + y2b2 = 1 according as x21a2 + y21b2 – 1 > 0, = or < 0.

6. If x2a2 + y2b2 = 1 is an ellipse, then its auxiliary circle is x2 + y2 = a2.

7. The equations x = a cos ф, y = b sin ф taken together are called the parametric equations of the ellipse x2a2 + y2b2 = 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.



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.




Share this page: What’s this?

Recent Articles

  1. What is Area in Maths? | Units to find Area | Conversion Table of Area

    Jul 17, 25 01:06 AM

    Concept of Area
    The amount of surface that a plane figure covers is called its area. It’s unit is square centimeters or square meters etc. A rectangle, a square, a triangle and a circle are all examples of closed pla…

    Read More

  2. Worksheet on Perimeter | Perimeter of Squares and Rectangle | Answers

    Jul 17, 25 12:40 AM

    Most and Least Perimeter
    Practice the questions given in the worksheet on perimeter. The questions are based on finding the perimeter of the triangle, perimeter of the square, perimeter of rectangle and word problems. I. Find…

    Read More

  3. Formation of Square and Rectangle | Construction of Square & Rectangle

    Jul 16, 25 11:46 PM

    Construction of a Square
    In formation of square and rectangle we will learn how to construct square and rectangle. Construction of a Square: We follow the method given below. Step I: We draw a line segment AB of the required…

    Read More

  4. Perimeter of a Figure | Perimeter of a Simple Closed Figure | Examples

    Jul 16, 25 02:33 AM

    Perimeter of a Figure
    Perimeter of a figure is explained here. Perimeter is the total length of the boundary of a closed figure. The perimeter of a simple closed figure is the sum of the measures of line-segments which hav…

    Read More

  5. Formation of Numbers | Smallest and Greatest Number| Number Formation

    Jul 15, 25 11:46 AM

    In formation of numbers we will learn the numbers having different numbers of digits. We know that: (i) Greatest number of one digit = 9,

    Read More