Science:Math Exam Resources/Courses/MATH105/April 2012/Question 08 (d)/Solution 1

If $z=e$ then

$e=e^{x^{2}+4y^{2}}$ and by taking the logarithm of both sides:

$\displaystyle 1=x^{2}+4y^{2}$

This is an ellipse.

The four easiest points are the intercepts.

If $x=0$ then $4y^{2}=1$ and $y=\pm 1/2$ are the y-intercepts. If $y=0$ then $x^{2}=1$ and $x=\pm 1$ are the x-intercepts. The sketch is below: