General Equation of an Ellipse

The general equation of an ellipse with major axis as the x- axis is

`((x-h)^2)/(a^2) + ((y-k)^2)/(b^2)= 1`

where (h,k) is the centre of the ellipse.

ellipse with major axis as x - axis

ellipse

The general equation of an ellipse with major axis as the y-axis is

`((x-h)^2)/(b^2) + ((y-k)^2)/(a^2)= 1`

ellipse with major axis as y - axis

ellipse

Lets solve some problems to find the general equation of ellipse.

Find the General Equation of an Ellipse - Examples

Ex : 1 Find the general equation of an ellipse whose vertices are (-1,4) and (-7,4) and eccentricity is `1/3` .

Sol : From the given data the major axis is parallel to x- axis.

`:.` The general equation form of an equation is `((x-h)^2)/(a^2) + ((y-k)^2)/(b^2) =1`

The centre is the midpoint of AA'

example 1 to find general equation

`:.` Cis ( `(-1-7)/2,(4+4)/2` ) = (-4,4)

Thus the equation becomes

`((x+4)^2)/(a^2) +( (y-4)^2)/(b^2) = 1`

We know that AA' = 2a = 6 `rArr` a=3

b2 = a2(1-e2) = 9( 1- `1/9` ) = 8

The required equation is `((x+4)^2)/9 + ((y-4)^2)/8 = 1`

Ex 2: Find the equation of the ellipse whose foci are (1,3) and (1,9) and eccentricity is `1/2`

Sol : From the given data the major axis is parallel to y-axis

`:.` the general equation is of the form

`((x-h)^2)/(b^2) + ((y-k)^2)/(a^2) = 1`

example 2 to find general equation

The centre of the ellipse is the midpoint of F1F2

`:.` C is ( `(1+1)/2 , (3+9)/2` ) = (1,6)

F1F2 = 2ae = 6

`rArr` ae = 3

But e = `1/2`       `:.` a = 6

b2 = a2(1-e2) = 36 ( 1-`1/4` ) = 27

Thus required equation is

`((x-1)^2)/(27) + ((y-6)^2)/36 = 1

I am planning to write more post on Equation of a Hyperbola and factoring polynomials of degree 3. Keep checking my blog.

Some more Examples to Find the General Equation of an Ellipse:

Ex 3: Find the equation of the ellipse  given that the centre is (4,-1) and one of its focus is (1,-1) and passing through (8,0)

Sol : From the given data since the major axis is parallel to the x axis , the equation is of the form

`((x-h)^2)/(a^2) + ((y-k)^2)/(b^2) =1`

The centre is C(h,k) is (4,-1).

example 3 to find general equation

`((x-4)^2)/(a^2) + ((y+1)^2)/(b^2) =1`

It passes through (8,0) `:. 16/(a^2) + 1/(b^2) = 1`

But CF1 = ae = 3

b2= a2(1-e2) = a2 - a2e2 = a2-9

(1) `rArr 16/(a^2) + 1/(a^2 -9) =1`

`rArr ` 16a2 - 144+a2 = a4 - 9a2

`rArr` a4 - 26a2 +144= 0

`rArr` a2 = 18 or a2 = 8

Case (i) a2 = 18 `rArr` b2 = a2 -9 = 18-9 = 9

Case (ii) a2 = 8  `rArr` b2 = 8-9 = -1 which is not possible

`:. ` a2 = 18 and b2 = 9

Thus the equation is `((x-4)^2)/18 + ((y+1)^2)/9 = 1`