Lesson Transform a general equation of an ellipse to the standard form by completing the square

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<H2>Transform a general equation of an ellipse to the standard form by completing the square</H2>

This lesson will show you by examples on how to transform a general equation of an ellipse to the standard equation by completing the squares.

<H3>Problem 1</H3>Transform a general equation of an ellipse &nbsp;{{{x^2+4y^2-4x - 12}}} = {{{0}}} to the standard equation and identify 
its center, the semi-axes, vertices, co-vertices, linear eccentricity and the foci.

<B>Solution</B>

<pre>
{{{x^2+4y^2-4x - 12}}} = {{{0}}}  ---->    (collect the quadratic and the linear terms with x and y in the left side; move the constant term to the right side)

{{{x^2-4x+4y^2}}} = {{{12}}}    ---->    (complete the squares for x and y separately)  --->

{{{(x^2 -4x) + 4y^2}}} = {{{12}}}   ---->    (group x-terms and y-terms separately)

{{{(x^2 -2*2x + 2^2) + 4y^2)}}} = {{{12 + 2^2}}}  --->  (add the necessary constant term {{{2^2}}} to x-terms to complete the square for x in the left side. 
                                     Add {{{2^2}}} to the right side accordingly.
                                     The y-term is just the completed square, so you do nothing with it)

{{{(x-2)^2 + 4y^2}}} = {{{16}}}     ---->    (write with completed squares and calculate the updated right side.<TABLE> 
  <TR>
  <TD> 
                                     Next divide both sides by the updated right side 16 = {{{4^2}}} )

{{{(x-2)^2/4^2}}} + {{{y^2/2^2}}} = 1     ---->    (You just got the ellipse equation in the standard form)

The center of the ellipse is the point  (2,0).

The major axis is y = 0 coinciding with x-axis.

The semi-major axis has the length of a = 4.

The semi-minor axis has the length of b = 2.   (Notice that a > b. The major axis is horizontal, 
                                                       and the ellipse is wider than tall).

Two vertices are  (2+4,0) = (6,0) and  (2-4,0) = (-2,0).

Two co-vertices are  (2,0-2) = (2,-2) and  (2,0+2) = (2,2).

The linear eccentricity is c = {{{sqrt(a^2-b^2)}}} = {{{sqrt(4^2-2^2)}}} = {{{sqrt(16-4)}}} = {{{sqrt(12)}}} = {{{2*sqrt(3)}}}.
  </TD>
   <TD>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{{{   drawing( 330, 330, -3.5, 7.5, -5.5, 5.5,
          circle(2, 0, 0.15), circle(2-2sqrt(3), 0, 0.15), circle(2+2*sqrt(3), 0, 0.15),

graph( 330, 330, -3.5, 7.5, -5.5, 5.5,
              2*sqrt(1-((x-2)^2/16)),
             -2*sqrt(1-((x-2)^2/16)))
)}}}

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Ellipse {{{(x-2)^2/16}}} + {{{y^2/4}}} = 1
  </TD>
  </TR>
</TABLE>The foci are  ({{{2-2*sqrt(3)}}},{{{0}}})  and  ({{{2+2*sqrt(3)}}},{{{0}}}).
</pre>

<H3>Problem 2</H3>Transform a general equation of an ellipse &nbsp;{{{9x^2+16y^2-126x+64y}}} = {{{71}}} to the standard equation and identify 
its center, the semi-axes, vertices, co-vertices, linear eccentricity and the foci.

<B>Solution</B>

<pre>
{{{9x^2+16y^2-126x+64y}}} = {{{71}}}     --->  (complete the squares for x and y separately)  --->

{{{9*(x^2-14x) + 16*(y^2+4y)}}} = {{{71}}}  --->  (collect the quadratic and the linear terms with x and y in the left side; keep the constant term in the right side)

{{{9*(x^2-2*7x + 7^2) + 16*(y^2+2*2y+2^2)}}} = {{{71 + 9*7^2+16*2^2}}}  --->  (add necessary constant terms to both sides to complete the squares in the left side)

{{{9*(x-7)^2 + 16*(y+2)^2}}} = {{{576}}}     --->   (write with completed squares and calculate the updated right side)<TABLE> 
  <TR>
  <TD> 
                                  ( Next divide both sides by the updated right side 576 = {{{24^2}}} )

{{{(x-7)^2/(24/3)^2}}} + {{{(y+2)^2/(24/4)^2}}} = 1,  or

{{{(x-7)^2/8^2}}} + {{{(y+2)^2/6^2}}} = 1.               (You just got the ellipse equation in the standard form)

The center of the ellipse is the point  (7,-2).

The major axis is y = -2 parallel to x-axis.

The semi-major axis has the length of a = 8.

The semi-minor axis has the length of b = 6.   (Notice that a > b. The major axis is horizontal, 
                                                       and the ellipse is wider than tall).

Two vertices are  (7+8,-2) = (15,-2) and  (7-8,-2) = (-1,-2).

Two co-vertices are  (7,-2+6) = (7,4) and  (7,-2-6) = (7,-8).

The linear eccentricity is c = {{{sqrt(a^2-b^2)}}} = {{{sqrt(8^2-6^2)}}} = {{{sqrt(64-36)}}} = {{{sqrt(28)}}} = {{{2*sqrt(7)}}}.
  </TD>
   <TD>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{{{   drawing( 360, 210, -4.5, 19.5, -8.5, 5.5,
          circle(7, -2, 0.25), circle(7-2sqrt(7), -2, 0.25), circle(7+2*sqrt(7), -2, 0.25),

graph( 360, 210, -4.5, 19.5, -8.5, 5.5,
             -2 + 6*sqrt(1-((x-7)^2/64)),
             -2 - 6*sqrt(1-((x-7)^2/64)))
)}}}

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Ellipse {{{(x-7)^2/8^2}}} + {{{(y+2)^2/6^2}}} = 1
  </TD>
  </TR>
</TABLE>The foci are  ({{{7+2*sqrt(7)}}},{{{-2}}})  and  ({{{7-2*sqrt(7)}}},{{{-2}}}).
</pre>

My other lessons on ellipses in this site are

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Quadratic-relations-and-conic-sections/Ellipse-definition--canonical-equation--characteristic-points-and-elements.lesson>Ellipse definition, canonical equation, characteristic points and elements</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Quadratic-relations-and-conic-sections/Ellipse-focal-property.lesson>Ellipse focal property</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Quadratic-relations-and-conic-sections/Tangen-lines-to-a-circle.lesson>Tangent lines and normal vectors to a circle</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Quadratic-relations-and-conic-sections/Tangent-lines-to-an-ellipse.lesson>Tangent lines and normal vectors to an ellipse</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Quadratic-relations-and-conic-sections/Optical-property-of-an-ellipse.lesson>Optical property of an ellipse</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Quadratic-relations-and-conic-sections/Optical-property-of-an-ellipse-revisited.lesson>Optical property of an ellipse revisited</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Quadratic-relations-and-conic-sections/Standard-equation-of-an-ellipse.lesson>Standard equation of an ellipse</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Quadratic-relations-and-conic-sections/Identify-elements-of-an-ellipse-given-by-its-standard-eqn.lesson>Identify elements of an ellipse given by its standard equation</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Quadratic-relations-and-conic-sections/Find-a-standard-equation-of-an-ellipse-given-by-its-elements.lesson>Find the standard equation of an ellipse given by its elements</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Quadratic-relations-and-conic-sections/General-equation-of-an-ellipse.lesson>General equation of an ellipse</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Quadratic-relations-and-conic-sections/Identify-vertices-co-vertices-foci-of-the-ellipse-given-by-an-equation.lesson>Identify elements of an ellipse given by its general equation</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Quadratic-relations-and-conic-sections/Standard-equation-of-a-circle.lesson>Standard equation of a circle</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Quadratic-relations-and-conic-sections/Find-the-equation-of-the-circle-given-by-its-center-and-tauching-a-given-line.lesson>Find the standard equation of a circle</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Quadratic-relations-and-conic-sections/General-equation-of-a-circle.lesson>General equation of a circle</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Quadratic-relations-and-conic-sections/Transform-general-eqn-of-a-circle-to-the-standard-form-by-completing-the-squares.lesson>Transform general equation of a circle to the standard form by completing the squares</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Quadratic-relations-and-conic-sections/Identify-elements-of-a-circle-given-by-its-general-equation.lesson>Identify elements of a circle given by its general equation</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Quadratic-relations-and-conic-sections/REVIEW-of-lessons-on-ellipses.lesson>OVERVIEW of lessons on ellipses</A>

Use this file/link &nbsp;<A HREF=https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-II - YOUR ONLINE TEXTBOOK</A>&nbsp; to navigate over all topics and lessons of the online textbook &nbsp;ALGEBRA-II.