Ellipse (Including the Circle Which Is a Special Case)

Summary

Conics

The 3 main types of conics are:

Parabola

Ellipse (including the circle which is a special case)

Hyperbola (including the rectangular hyperbola which is a special case)

General Equations of Conics:

Shape / Name / General Equation / Description / What does the equation tell us?

/ Parabola / y=ax2+bx+c
or
y2=ax+b / Only one squared term (x or y). / y-intercept is at c.
x-intercept at -ba
/ Circle / x2+y2=r2 / x and y terms both squared, and added. Coefficients of x2 and y2 are equal and there is no xy term. (special ellipse) / To find the radius make the coefficients of x2 and y2 equal 1.
Centre (0,0) radius= r

/ Ellipse / x2a2+y2b2=1 / x and y terms both squared, and added / Intercepts x-axis at ±a and y-axis at ±b (notice that when a=b it’s a circle)

/ Hyperbola / x2a2-y2b2=1 / x and y terms both squared, then subtracted / The diagonals of the rectangle (formed with length from –a to a and height from –b to b) are asymptotes for the hyperbola.

/ Hyperbola / y2a2-x2b2=1 / x and y terms both squared, then subtracted / The hyperbola is still formed using the rectangle. y2 first turns it the other way.

N.b. a and b 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. 4x29 becomes x294 (using the rules for dividing fractions)).

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?
/ Circle / (x-a)2+y-b2=r2 / Notice the extra a and b. To find the centre and the radius make the coefficients of x2 and y2 equal 1.
Centre at (a,b) radius = r

/ Ellipse / x-x02a2+y-y02b2=1 / Notice the extra x0 and y0
Centre at (x0,y0) radius is a along the x-axis and b along the y-axis

/ Hyperbola / x-x02a2-y-y02b2=1 / Notice the extra x0 and y0
Centre at (x0,y0)
The diagonals of the rectangle (with length 2a and height 2b) 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
9x2+4y2=36 / We need the equation to =1
(÷36) x24+y29=1 / Ellipse
Centre at (0,0) radius is 2 along the x-axis and 3 along the y-axis
x2-6x+y2+8y=-16 / Complete the square for x and y.
x-32-32+y+42-42=-16
x-32+y+42=9 / Circle
Centre at (3,-4)
radius = 3
x2-9y2-4x+18y=14 / Complete the square and rearrange
x2-4x-9y2-2y=14
x-22-4-9y-12-1=14
x-22-4-9y-12+9=14
x-22-9y-12=9
(÷9) x-229-y-121=1 / Hyperbola
Centre at (2,-1)
The diagonals of the rectangle (length -3 and +3 from the centre and height -1 and +1 from the centre) are asymptotes for the hyperbola.