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:


x2 + y2 = 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 = ±   _____
Ö1 - x2
 

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:


x2
36
 + y2
16
 = 1     (2)

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


y = ±   æ
 ú
Ö
16( 1 - x2
36
 )
 
 = ±4   æ
 ú
Ö
1 - ( x2
36
 )
 

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

Now, as I understood that in the general equation:


x2
a2
 + y2
b2
 = 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:


x2
72
 + y2
42
 = 1     (4)

x2
52
 + y2
42
 = 1     (5)

x2
42
 + y2
42
 = 1     (6)

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


x2
32
 + y2
42
 = 1     (7)

x2
22
 + y2
42
 = 1     (8)

x2
12
 + y2
42
 = 1     (9)

x2
62
 + y2
32
 = 1     (10)

x2
72
 + y2
12
 = 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.