SOLUTION: Ellipse Hello, would you please help me? Express 4y^2 + 9x^2 - 24y - 72x + 144 = 0 to standard equation, then graph the ellipse. Thank you very much. It would be a big help

Algebra ->  Quadratic-relations-and-conic-sections -> SOLUTION: Ellipse Hello, would you please help me? Express 4y^2 + 9x^2 - 24y - 72x + 144 = 0 to standard equation, then graph the ellipse. Thank you very much. It would be a big help      Log On


   

Question 1089712: Ellipse
Hello, would you please help me?
Express 4y^2 + 9x^2 - 24y - 72x + 144 = 0
to standard equation, then graph the ellipse.
Thank you very much. It would be a big help.
Answer by ikleyn(52803) About Me  (Show Source):
You can put this solution on YOUR website!
.
Express 4y^2 + 9x^2 - 24y - 72x + 144 = 0
to standard equation, then graph the ellipse.
~~~~~~~~~~~~~~~~~~~~~~ 

The method is completing the squares separately for x-terms and y-terms.
I will do it step by step - trace my steps very attentively.


4y^2 + 9x^2 - 24y - 72x + 144 = 0

4y^2 + 9x^2 - 24y - 72x = -144

(9x^2 - 72x) + (4y^2 - 24y) = -144

(9x^2 - 2*3*12x + 12^2) + (4y^2 -2*2*6y + 6^2) = -144 + 12^2 + 6^2

(3x-12)^2 + (2y-6)^2 = -144 + 144 + 36 = 36   (*)


Now divide both sides of the equation (*) by 36. You will get

%283x-12%29%5E2%2F36 + %282y-6%29%5E2%2F36 = 1.


Make couple of steps further

%289%2A%28x+-+4%29%5E2%29%2F36 + %284%2A%28y-3%29%5E2%29%2F36 = 1

 
%28%28x+-+4%29%5E2%29%2F4 + %28%28y-3%29%5E2%29%2F9 = 1


%28%28x+-+4%29%5E2%29%2F2%5E2 + %28%28y-3%29%5E2%29%2F3%5E2 = 1    (**)


Equation (**) is the standard form of an ellipse equation.


The plot is shown below:


        


            Ellipse %28x-4%29%5E2%2F2%5E2 + %28y-3%29%5E2%2F3%5E2 = 1


The center of the ellipse is at the point (4,3).
The major axis is vertical.  The ellipse is taller than wide.
The major semi-axis is 3 units long.
The minor semi-axis is 2 units long.
The linear eccentricity is sqrt%283%5E2-2%5E2%29 = sqrt%285%29.

Solved.


There are lessons in this site that explain everything about ellipses, their equations and elements,
about finding and identifying the ellipse elements via equations and vice versa,
as well as transforming general equations of ellipses to their standard forms.

These lessons are
    - Ellipse definition, canonical equation, characteristic points and elements

    - Standard equation of an ellipse
    - Identify elements of an ellipse given by its standard equation
    - Find the standard equation of an ellipse given by its elements

    - General equation of an ellipse
    - Transform a general equation of an ellipse to the standard form by completing the square
    - Identify elements of an ellipse given by its general equation

and you will not find the better place in the Internet to learn this subject.


Also,  you have this free of charge online textbook in ALGEBRA-II in this site
    ALGEBRA-II - YOUR ONLINE TEXTBOOK.

The referred lessons are the part of this online textbook under the topic
"Conic sections: Ellipses. Definition, major elements and properties. Solved problems".