Well done Ryan and Prateek from Riccarton High School, Christchurch, New Zealand and Andrei from School 205, Bucharest, Romania for your solutions.

Ryan noticed that the formula x ² / 36 + y ² / 16 = 1 gives a large ellipse crossing through points 6 and -6 on the x axis and points 4 and -4 on the y axis. From this he observed that the formula contains (x ² / 36) and 36 is a square of 6 and -6. When x takes these values then y=0. This also works for the part of the formula (y ² / 16) as 16 is a square of 4 and -4. When y takes these values then x=0.

Andrei explained how he found the other equations as follows.

I represented first the curve:

x^{2} + y^{2} = 1 (1)

For this, I observed that both x and y could have values between -1 and +1. I consider x as the independent variable, and from the eq. (1) I determined y:

y = \pm \sqrt{1 - x^{2}}

I gave values to x, from -1 to 1, step 0.1, and I calculated y. I had to calculate two sets of values for y, one corresponding to the plus sign, the other to the minus sign. Then I plotted y as a function of x for both, and I obtained the circle in the middle.

For the curve:

\frac{x^{2}}{36} + \frac{y^{2}}{16} = 1 (2)

I considered again x as the independent variable. It varies between -6 to 6. The equation for y is:

y = \pm \sqrt{16 (1 - \frac{x^{2}}{36})} = \pm 4 \sqrt{1 - (\frac{x^{2}}{36})}

It is visible even from the equation that y varies between -4 and 4.

Now, as I understood that in the general equation:

\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (3)

x varies between -a and a, and y between -b and b, I drew all other curves in the same manner. The equations of the other 8 graphs are:

\frac{x^{2}}{7^{2}} + \frac{y^{2}}{4^{2}} = 1 (4)

\frac{x^{2}}{5^{2}} + \frac{y^{2}}{4^{2}} = 1 (5)

\frac{x^{2}}{4^{2}} + \frac{y^{2}}{4^{2}} = 1 (6)

or , i.e. a circle of radius 4.

\frac{x^{2}}{3^{2}} + \frac{y^{2}}{4^{2}} = 1 (7)

\frac{x^{2}}{2^{2}} + \frac{y^{2}}{4^{2}} = 1 (8)

\frac{x^{2}}{1^{2}} + \frac{y^{2}}{4^{2}} = 1 (9)

\frac{x^{2}}{6^{2}} + \frac{y^{2}}{3^{2}} = 1 (10)

\frac{x^{2}}{7^{2}} + \frac{y^{2}}{1^{2}} = 1 (11)

All these ellipses are symmetrical about both x and y axes because by changing x to -x and/or y to -y the equation (3) doesn't change.