Academic Skills Advice
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
𝒚 = 𝒂𝒙𝟐 + 𝒃𝒙 + 𝒄 Only one squared term
or
𝒚𝟐 = 𝒂𝒙 + 𝒃
𝒙 𝟐 + 𝒚𝟐 = 𝒓 𝟐
𝒙𝟐
Ellipse
𝒂𝟐
+
𝒚𝟐
𝒃𝟐
=𝟏
𝑏
𝑥-intercept at − 𝑎
𝑥 and 𝑦 terms both
squared, and added.
Coefficients of 𝑥 2 and
𝑦 2 are equal and there
is no 𝑥𝑦 term. (special
ellipse)
To find the radius
make the coefficients
of 𝑥 2 and 𝑦 2 equal 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)
The diagonals of the
rectangle (formed
with length from –a
to a and height from
–b to b) are
asymptotes for the
hyperbola.
asymptotes
𝒙𝟐
b
a
Hyperbola
a
b
Hyperbola
𝑦-intercept is at c.
(𝑥 or 𝑦).
Parabola
Circle
What does the
equation tell us?
𝒂𝟐
−
𝒚𝟐
𝒃𝟐
𝒚𝟐
𝒙𝟐
𝒂
𝒃𝟐
𝟐 −
=𝟏
𝑥 and 𝑦 terms both
squared, then
subtracted
=𝟏
𝑥 and 𝑦 terms both
squared, then
subtracted
Centre (0,0) radius= 𝑟
The hyperbola is still
formed using the
rectangle. 𝑦 2 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.
4𝑥 2
9
becomes
𝑥2
9
4
( )
(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.
© H Jackson 2011 / 2015 / ACADEMIC SKILLS
1
Equations of (translated) conics:
Shape
Name
General Equation
What does the equation tell us?
(𝒙 − 𝒂)𝟐 + (𝒚 − 𝒃)𝟐 = 𝒓𝟐
Notice the extra 𝒂 and 𝒃. To find
the centre and the radius make the
coefficients of 𝑥 2 and 𝑦 2 equal 1.
Circle
Centre at (𝒂,𝒃) radius = 𝑟
Notice the extra 𝒙𝟎 and 𝒚𝟎
(𝒙−𝒙𝟎 )𝟐
Ellipse
𝒂𝟐
+
(𝒚−𝒚𝟎 )𝟐
𝒃𝟐
Centre at (𝒙𝟎 , 𝒚𝟎 ) radius is 𝑎 along
the 𝑥-axis and 𝑏 along the 𝑦-axis
=𝟏
Notice the extra 𝒙𝟎 and 𝒚𝟎
asymptotes
(𝒙−𝒙𝟎 )𝟐
2𝑎
Hyperbola
2𝑏
𝒂𝟐
−
(𝒚−𝒚𝟎 )𝟐
Centre at (𝒙𝟎 , 𝒚𝟎 )
=𝟏
𝒃𝟐
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
We need the equation to =1
Conic
Ellipse
9𝑥 2 + 4𝑦 2 = 36
𝒙𝟐
(÷ 36)
𝟒
+
𝒚𝟐
𝟗
Centre at (0,0) radius
is 2 along the 𝑥-axis
and 3 along the 𝑦-axis
=𝟏
Complete the square for 𝑥 and 𝑦.
2
Circle
2
𝑥 − 6𝑥 + 𝑦 + 8𝑦 = −16
(𝑥 − 3)2 − 32 + (𝑦 + 4)2 − 42 = −16
𝟐
𝟐
(𝒙 − 𝟑) + (𝒚 + 𝟒) = 𝟗
𝑥 2 − 9𝑦 2 − 4𝑥 + 18𝑦 = 14
Complete the square and rearrange
𝑥 2 − 4𝑥 − 9(𝑦 2 − 2𝑦) = 14
(𝑥 − 2)2 − 4 − 9((𝑦 − 1)2 − 1) = 14
(𝑥 − 2)2 − 4 − 9(𝑦 − 1)2 + 9 = 14
(𝑥 − 2)2 − 9(𝑦 − 1)2 = 9
(÷ 9)
© H Jackson 2011 / 2015 / ACADEMIC SKILLS
(𝒙−𝟐)𝟐
𝟗
−
(𝒚−𝟏)𝟐
𝟏
=𝟏
Centre at (3,-4)
radius = 3
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.
2