Question 1092125
.
Rewrite the relation as a conic in standard form, then sketch the graph of the relation. 

{{{ 2x=sqrt( 8y-y^2 ) }}}
~~~~~~~~~~~~~~~~~~


<pre>
2x = {{{sqrt( 8y-y^2 ) }}}   ====>  square both sides  ====>

{{{4x^2}}} = {{{8y - y^2}}} ====>  collect all retms in the left side  ====>

{{{4x^2 + y^2 - 8y}}} = 0  ====> Complete the square in the group of y-terms  ====>

{{{4x^2}}} + {{{(y-4)^2}}} = 16  ====>  Divide both sides by 16  ====>

{{{x^2/2^2}}} + {{{(y-4)^2/4^2}}} = 1.


You got the standard equation of an ellipse.


The major axis is parallel to y-axis and is vertical.
The minor axis is parallel to x-axis and is horizontal.

The major semi-axis has the length of 4 units.
The minor semi-axis has the length of 2 units.

The ellipse is taller than wide.

The center is at the point (x,y) = (0,2).

The linear eccentricity is {{{sqrt(2^4 - 2^2)}}} = {{{sqt(12)}}} = {{{2*sqrt(3)}}}.

The foci are at the points ({{{0}}},{{{2-2sqrt(3)}}})  and  ({{{0}}},{{{2+2sqrt(3)}}})
</pre>


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

      graph( 330, 330, -5.5, 5.5, -3.5, 7.5,
              2 + 4*sqrt(1-(x^2/4)),
              2 -4*sqrt(1-(x^2/4)))
)}}}


&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Ellipse {{{x^2/4}}} + {{{(y-2)^2/16}}} = 1


----------------
See the lessons in this site

&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=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/Transform-general-eqn-of-an-ellipse-to-the-standard-form-by-completing-the-square.lesson>Transform a general equation of an ellipse to the standard form by completing the square</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>



Also, &nbsp;you have this free of charge online textbook in ALGEBRA-II in this site

&nbsp;&nbsp;&nbsp;&nbsp;<A HREF=https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-II - YOUR ONLINE TEXTBOOK</A>.


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



Save the link to this textbook together with its description


Free of charge online textbook in ALGEBRA-II
https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson


into your archive and use when it is needed.