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 + 3y2 = 12
Solution :
4x2 + 3y2 = 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[x2 + 2x(1) + 12 - 12] + 9[y2 - 2y(2) + 22 - 22] - 12 = 0
6[(x + 1)2 - 1] + 9[(y - 2)2 - 4] - 12 = 0
6(x + 1)2 - 6 + 9(y - 2)2 - 36 - 12 = 0
6(x + 1)2 + 9(y - 2)2 - 54 = 0
6(x + 1)2 + 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[x2 - 2x(1) + 12 - 12] + 4[y2+ 2y(4) + 42 - 42] - 44 = 0
36[(x - 1)2- 1] + 4[(y + 4)2- 16] - 44 = 0
36(x - 1)2 - 36 + 4(y + 4)2 - 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
Kindly mail your feedback to v4formath@gmail.com
We always appreciate your feedback.
WORD PROBLEMS
Word problems on simple equations
Word problems on linear equations
Word problems on quadratic equations
Area and perimeter word problems
Word problems on direct variation and inverse variation
Word problems on comparing rates
Converting customary units word problems
Converting metric units word problems
Word problems on simple interest
Word problems on compound interest
Word problems on types of angles
Complementary and supplementary angles word problems
Markup and markdown word problems
Word problems on mixed fractions
One step equation word problems
Linear inequalities word problems
Ratio and proportion word problems
Word problems on sets and Venn diagrams
Pythagorean theorem word problems
Percent of a number word problems
Word problems on constant speed
Word problems on average speed
Word problems on sum of the angles of a triangle is 180 degree
OTHER TOPICS
Time, speed and distance shortcuts
Ratio and proportion shortcuts
Domain and range of rational functions
Domain and range of rational functions with holes
Graphing rational functions with holes
Converting repeating decimals in to fractions
Decimal representation of rational numbers
Finding square root using long division
L.C.M method to solve time and work problems
Translating the word problems in to algebraic expressions
Remainder when 2 power 256 is divided by 17
Remainder when 17 power 23 is divided by 16
Sum of all three digit numbers divisible by 6
Sum of all three digit numbers divisible by 7
Sum of all three digit numbers divisible by 8
Sum of all three digit numbers formed using 1, 3, 4
Sum of all three four digit numbers formed with non zero digits
Sum of all three four digit numbers formed using 0, 1, 2, 3
Sum of all three four digit numbers formed using 1, 2, 5, 6
©All rights reserved. onlinemath4all.com
May 23, 22 01:59 AM
Linear vs Exponential Growth - Concept - Examples
May 23, 22 01:42 AM
Exponential vs Linear Growth Worksheet
May 23, 22 01:34 AM
SAT Math Questions on Exponential vs Linear Growth