Question 1073654
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please help me Find the center, vertices, foci and endpoints with the minor axis of the ellipse of 2x^2+3y^2=18
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<pre>
2x^2+3y^2 = 18   --->  (divide both sides by 18. You will get)  --->

{{{x^2/9 + y^2/6}}} = 1  --->

{{{x^2/3^2 + y^2/(sqrt(6))^2}}} = 1.


It is the canonical equation of the ellipse with the center at the origin (x,y) = (0,0), 
the major semi-axis a = 3, the minor semi-axis b = {{{sqrt(6)}}}.

Vertices are (3,0), (-3,0), ({{{0}}}, {{{sqrt(6)}}}),  ({{{0}}},{{{-sqrt(6)}}}),

The linear eccentricity of the ellipse is the number {{{c}}} = {{{sqrt(a^2-b^2)}}} = {{{sqrt(3^2-(sqrt(6))^2)}}} = {{{sqrt(3)}}}.

The foci are the points (c,0) = ({{{sqrt(3)}}},{{{0}}}) and (-c,0) = ({{{-sqrt(3)}}},{{{0}}}).
</pre>


{{{drawing( 322, 240, -4.0, 4.1, -3.0, 3.0,
            grid(1),
            ellipse(0.0, 0.0, 6.0, 2*sqrt(6))
)}}}


<B>Figure</B>. &nbsp;The ellipse &nbsp;&nbsp;{{{x^2/3^2 + y^2/(sqrt(6))^2}}} = 1.


See the lessons

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://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/Identify-vertices-co-vertices-foci-of-the-ellipse-given-by-an-equation.lesson>Identify the vertices, co-vertices and foci of the ellipse given by an 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 a standard equation of an ellipse given by its elements</A>

in this site.


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>".