# MAJOR AXIS AND MINOR AXIS OF ELLIPSE

Standard equation of an ellipse (symmetric about x-axis) :

Here, the center of the ellipse is (h, k).

When the center of the ellipse is origin (0, 0), then the above equation becomes as shown below.

Here a > b. Major Axis :

The line segment AA′ is called the major axis and the length of the major axis is 2a. The equation of the major axis is y = 0.

Minor Axis :

The line segment BB′ is called the minor axis and the length of minor axis is 2b. Equation of the minor axis is x = 0.

Standard equation of an ellipse (symmetric about y-axis) :

Here, the center of the ellipse is (h, k).

When the center of the ellipse is origin (0, 0), then the above equation becomes as shown below.

Here a > b. Length of the major axis = AA' = 2a.

Equation of the major axis is x = 0.

Length of the minor axis = BB' = 2b.

Equation of the minor axis is y = 0.

Note that the length of major axis is always greater than minor axis.

Find the equations and lengths of major and minor axes of the following ellipses.

Example 1 :

Solution :

The major axis is along x-axis and the minor axis is along y-axis.

Equation of major axis : y = 0.

Equation of minor axis : x = 0.

a2 = 9 ----> a = 3

b2 = 4 ----> b = 2

Length of major axis :

= 2a

= 2(3)

= 6

Length of minor axis :

= 2b

= 2(2)

= 4

Example 2 :

4x2 + 3y= 12

Solution :

4x2 + 3y= 12

Divide both sides by 12.

The major axis is along y-axis and the minor axis is along x-axis.

Equation of major axis : x = 0.

Equation of minor axis : y = 0.

a2 = 4 ----> a = 2

b2 = 3 ----> b = √3

Length of major axis :

= 2a

= 2(2)

= 4

Length of minor axis :

= 2b

= 2√3

Example 3 :

6x2 + 9y2 + 12x - 36y - 12 = 0

Solution :

The given equation of ellipse is not in standard form. Convert it to standard form.

6x2 + 9y2 + 12x - 36y - 12 = 0

6x2 + 12x + 9y2 - 36y - 12 = 0

6(x2 + 2x) + 9(y2 - 4y) - 12 = 0

6[x+ 2x(1) + 1- 12+ 9[y- 2y(2) + 2- 22- 12 = 0

6[(x + 1)- 1] + 9[(y - 2)- 4] - 12 = 0

6(x + 1)- 6 + 9(y - 2)- 36 - 12 = 0

6(x + 1)+ 9(y - 2)2 - 54 = 0

6(x + 1)+ 9(y - 2)2 = 54

Divide both sides by 54.

Let X = x - 1 and Y = y - 2.

Clearly the major axis is along X-axis and the minor axis is along Y-axis.

Equation of the major axis :

Y = 0

y - 2 = 0

y = 2

Equation of the minor axis :

X = 0

x + 1 = 0

x = -1

Here,

a2 = 9 ----> a = 3

b2 = 6 ----> b = 6

Length of major axis :

= 2a

= 2(3)

= 6

Length of minor axis :

= 2b

= 2√6

Example 4 :

36x2 - 72x + 4y2 + 32y - 44 = 0

Solution :

The given equation of ellipse is not in standard form. Convert it to standard form.

36x2 - 72x + 4y2 + 32y - 44 = 0

36(x2 - 2x) + 4(y2 + 8y) - 44 = 0

36[x- 2x(1) + 1- 12+ 4[y2+ 2y(4) + 4- 42- 44 = 0

36[(x - 1)2- 1] + 4[(y + 4)2- 16] - 44 = 0

36(x - 1)- 36 + 4(y + 4)- 64 - 44 = 0

36(x - 1)2+ 4(y + 4)2 - 144 = 0

36(x - 1)2+ 4(y + 4)2 = 144

Divide both sides by 144.

Let X = x - 1 and Y = y + 4.

Clearly the major axis is along Y-axis and the minor axis is along X-axis.

Equation of the major axis :

X = 0

x - 1 = 0

x = 1

Equation of the minor axis :

Y = 0

y + 4 = 0

y = -4

Here,

a2 = 36 ----> a = 6

b2 = 4 ----> b = 2

Length of major axis :

= 2a

= 2(6)

= 12

Length of minor axis :

= 2b

= 2(2)

= 4

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