Circles, ellipses and hyperbolas

For standard curves like circles, ellipses and hyperbolas, you need to recognise them from the equation, then apply SIA (usually just Intercepts & Asymptotes) to complete the picture.

Equation of a circle look like x^2 + y^2 = r^2. Alternatively, if you divide throughout by r^2, we get (x^2/r^2) + (y^2/r^2) = 1.

Compare this to that of an ellipse: (x^2/a^2) + (y^2/b^2) = 1. Thus, the circle is a special case of the ellipse when a = b.

Note that so far, we are only talking about the circle/ellipse having its centre at the origin. If the numerators in the above equations are changed then the centre would change too. For example, if we have (x - 3)^2 + (y + 5)^2 = 36, then this circle's centre is at (3, -5).

To find the x and y coordinates of the centre, just equate the expression inside the respective bracket to zero. Same thing applies for ellipses.

When the centre is not the origin, you need to take note whether the origin is inside, on, or outside the circle/ellipse. This can be easily found by subsituting x = 0, into the equation.

1) If y gives no answer or only 1 non-zero value, then the origin lies outside of the circle/ellipse.

2) If one of the values of y is zero, then the origin lies on the perimeter of the circle/ellipse.

3) If y gives 2 non-zero answers, then the origin lies inside the circle/ellipse.

Hyperbolas take the form of (x^2/a^2) - (y^2/b^2) = 1 or (y^2/b^2) - (x^2/a^2) = 1. The only difference from the ellipse equation is that you have a minus instead of a plus.

Again, to complete the picture, you need to find the intercepts and asymptotes from theory. Alternatively, just remember the oblique asymptotes are always y = +(bx/a) and y = -(bx/a).