Ellipse (Including the Circle Which Is a Special Case)

1) Conics Summary Conics are curves that are produced when a double cone is intersected by a plane. The 3 main types of conics are: Parabola Ellipse (including the circle which is a special case) Hyperbola

General Equations of Conics: Shape Name General Equation Description What does the equation tell us?

Only one squared -intercept is at c. Parabola term ( or ). or -intercept at

and terms both To find the radius squared, and make the coefficients added. of and equal 1. Circle Coefficients of and are equal and there is no Centre (0,0) radius= term. (special ellipse)

and terms both Intercepts -axis at squared, and and -axis at (notice Ellipse added that when it’s a circle)

The diagonals of the and terms both rectangle (formed squared, then with length from –a Hyperbola subtracted to a and height from –b to b) are asymptotes for the hyperbola.

The hyperbola is still and terms both formed using the squared, then rectangle. first turns Hyperbola subtracted it the other way.

N.b. and could be fractions. Remember that if this is the case with the ellipse or the hyperbola you may need to turn the fraction “upside down” to send it to the bottom, (e.g. becomes (using the rules for dividing fractions)).

H Jackson 2010 & 2012 / Academic Skills 1 Sometimes the centre of the circle, ellipse or hyperbola is not on the origin – this can easily be seen from the equation.

Equations of (translated) conics: Shape Name General Equation What does the equation tell us? Notice the extra and . To find the centre and the radius make the coefficients of and equal 1. Circle Centre at (,) radius =

Notice the extra and

Centre at () radius is along the -axis Ellipse and along the -axis

Notice the extra and

Centre at () Hyperbola The diagonals of the rectangle (with length and height ) are asymptotes for the hyperbola.

N.b. The “rectangular hyperbola” is a special case where the asymptotes are perpendicular to each other (i.e. a=b, so a square is formed)

Some equations may need rearranging to help decide on the type of conic they are.

Examples:

Starting Equation Rearrange Conic We need the equation to =1 Ellipse Centre at radius is along the -axis and along the -axis Complete the square for and . Circle

Centre at (,-4) radius = Complete the square and rearrange Hyperbola Centre at

The diagonals of the rectangle (length -3 and +3 from the centre and height and +1 from the centre) are asymptotes for the hyperbola.

H Jackson 2010 & 2012 / Academic Skills 2