Scroll down to the bottom to view the interactive graph.

The signs of $$a$$, $$b$$ and $$c$$ determine the shape of the quadric surface defined by $ax^2 + by^2 + cz^2 = 1.$ For example if we keep $$a$$ and $$b$$ positive, then a negative $$c$$ gives a hyperboloid of one sheet with the centre axis along the $$z$$-axis, while positive $$c$$ gives an ellipsoid (or a sphere if $$a = b = c$$). The following figure shows the transition from hyperboloid to ellipsoid by varying $$c$$.

Can you determine the values of $$a$$, $$b$$ and $$c$$ that give two parallel planes? You may explore using the interactive graph provided at the bottom of the page.

To view the interactive graph:

1. Make sure you have the latest version of Java 7 installed in your computer. Tablets and smartphones are not supported.