Suppose that a student only has the theoretical knowledge on circles yet he's attempting a problem to include an ellipse. Is it possible to alter the coordinate axes i.e. $x$ as $x+a$ and if so are there any limitations for doing so (i.e.is it possible to have $x^n$ instead of $x$,..)
If its possible,then are there relationships between conic sections that follow a certain pattern.
I feel like that its possible for an ellipse,but i'm not sure if I'm violating any basic principles.
If possible please elaborate on the subject.
Please hold your enthusiasm until you learn how the equations for ellipses are modified from circles. Or better to say how they are generalized so that circles form their special cases.
When axes of ellipses are parallel to coordinate axes, two axes are identified, constants are added or subtracted, ... to change their relative proportion.
$$ \dfrac {(x)^2}{a^2}+ \dfrac {(y)^2}{a^2}=1 $$
can be changed to
$$ \dfrac {(x)^2}{a^2}+ \dfrac {(y)^2}{b^2}=1 $$
where $b>a.$