Question 1089223
.
<pre>
The graph of 2y^4 -x^2 +11 =0 is symmetric with respect which of the following?
 1. The x axis 
 2. The y axis 
 3. The origin 
A) only 1
B) only 2
C) only 3
D) 1, 2, and 3


{{{graph( 330, 330, -10.5, 10.5, -5.5, 5.5,
          sqrt(sqrt((x^2-11)/2)), -sqrt(sqrt((x^2-11)/2))
)}}}


Plot y1 = {{{root(4,(x^2-11)/2))}}} (red)  and y2 = {{{-root(4,(x^2-11)/2))}}} (green).
</pre>


&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;The correct answer is &nbsp;OPTION &nbsp;D).



<pre>
0.  Let the point (x,y) belongs to the curve.

    It means that its coordinates satisfy the equation 

    2y^4 -x^2 +11 =0.     (1)



1.  Then the point (-x,y) also belongs to the curve.

    Indeed, then 2y^4 - (-x)^2 + 11 = 2y^4 - x^2 + 11 = 0 due to (1).

    It means that the symmetry  (x,y) --> (-x,y)  relative to the axis "Y" is in place.



2.  Also, then the point (x,-y) belongs to the curve.

    Indeed, then 2(-y)^4 - x^2 + 11 = 2y^4 - x^2 + 11 = 0 due to (1).

    It means that the symmetry  (x,y) --> (x,-y)  relative to the axis "X" is in place.



3.  Finally, then the point (-x,-y) belongs to the curve.

    Indeed, then 2(-y)^4 - (-x)^2 + 11 = 2y^4 - x^2 + 11 = 0 due to (1).

    It means that the symmetry  (x,y) --> (-x,-y)  relative to the origin is in place.
</pre>


Thus my statement is &nbsp;<U>PROVED</U>&nbsp; and the solution is &nbsp;<U>COMPLETED</U>.



&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<U>Proved</U>. &nbsp;&nbsp;<U>Solved</U>. &nbsp;&nbsp;<U>And completed</U>. 




The other's tutor solution is &nbsp;&nbsp;<U>WRONG</U> &nbsp;&nbsp;(his approach is &nbsp;&nbsp;<U>WRONG</U>&nbsp;&nbsp; and his answer is &nbsp;&nbsp;<U>UNCOMPLETED</U>).