New Zealand
Level 8 - NCEA Level 3

# Identifying and graphing conic sections

Lesson

The circle, ellipse, parabola and hyperbola are all types of conic sections. When a plane intersects a cone at different angles it will produces one of the four curves depending on the slope of the cone and of the plane.

 The four conic sections found by intersecting a cone.

The standard form(s) of each equation (for both the horizontal and vertical orientations) and the important graphing features for each conic section are presented in the table below:

Conic section Equation Graph Characteristics
Ellipse

$\frac{\left(x-h\right)^2}{a^2}+\frac{\left(y-k\right)^2}{b^2}=1$(xh)2a2+(yk)2b2=1

• Centre: $\left(h,k\right)$(h,k)
• Vertices: $\left(h\pm a,k\right)$(h±a,k)
• Co-vertices: $\left(h,k\pm b\right)$(h,k±b)
$\frac{\left(x-h\right)^2}{b^2}+\frac{\left(y-k\right)^2}{a^2}=1$(xh)2b2+(yk)2a2=1
• Centre: $\left(h,k\right)$(h,k)
• Co-vertices: $\left(h\pm b,k\right)$(h±b,k)
• Vertices: $\left(h,k\pm a\right)$(h,k±a)
Circle $\left(x-h\right)^2+\left(y-k\right)^2=r^2$(xh)2+(yk)2=r2
• Centre: $\left(h,k\right)$(h,k)
• Radius: $r$r
Hyperbola

$\frac{\left(x-h\right)^2}{a^2}-\frac{\left(y-k\right)^2}{b^2}=1$(xh)2a2(yk)2b2=1

• Centre: $\left(h,k\right)$(h,k)
• Vertices: $\left(h\pm a,k\right)$(h±a,k)
• Asymptotes: $y=\pm\frac{b}{a}\left(x-h\right)+k$y=±ba(xh)+k
$\frac{\left(y-k\right)^2}{a^2}-\frac{\left(x-h\right)^2}{b^2}=1$(yk)2a2(xh)2b2=1

• Centre: $\left(h,k\right)$(h,k)
• Vertices: $\left(h,k\pm a\right)$(h,k±a)
• Asymptotes: $y=\pm\frac{a}{b}\left(x-h\right)+k$y=±ab(xh)+k
Parabola

$y=a\left(x-h\right)^2+k$y=a(xh)2+k

• Vertex: $\left(h,k\right)$(h,k)
• $a$a determines concavity
• $y$y-intercept: when $x=0$x=0
• $x$x-intercepts: when $y=0$y=0
$x=a\left(y-k\right)^2+h$x=a(yk)2+h
• Vertex:$\left(h,k\right)$(h,k)
• $a$a determines concavity
• $y$y-intercepts: when $x=0$x=0
• $x$x-intercept: when $y=0$y=0

### The general conic form

Each conic section has its own standard form that makes it easiest to read off its characteristics. There is, however, a general form for all conic sections:

$Ax^2+By^2+Cx+Dy+E=0$Ax2+By2+Cx+Dy+E=0,

where $A$A, $B$B, $C$C, $D$D and $E$E are real numbers. All conic sections can be represented by this equation, but not all choices of $A$A, $B$B, $C$C, $D$D and $E$E will result in a conic section - as we will see below. This general form is useful for classifying which type of conic section a particular equation represents.

If an equation in this form does represent a conic section, we can determine which one it will be by looking at the values of $A$A and $B$B. The table below outlines this:

Conic

Value of $A$A and $B$B

Parabola

$A=0$A=0 or $B=0$B=0

Ellipse

$A$A and $B$B have the same sign, that is, $A\times B>0$A×B>0

Circle

$A$A = $B$B
Hyperbola $A$A and $B$B have the opposite sign, that is, $A\times B<0$A×B<0

To graph a conic in this form it is easiest to rearrange the equation (which often involves completing the square) into a form where it is easier to find its identifying features (centre, vertices, radius etc.).

Not always a conic section

Below are examples of when $Ax^2+By^2+Cx+Dy+E=0$Ax2+By2+Cx+Dy+E=0 does not represent a conic section:

• Line: $A=B=0$A=B=0 describes a straight line (if $C$C and $D$D are non-zero).
• No solution: $A=B=E=1,C=D=0$A=B=E=1,C=D=0 gives us $x^2+y^2+1=0$x2+y2+1=0 or $x^2+y^2=-1$x2+y2=1. This look likes it describes a circle - however it would have a radius of $\sqrt{-1}$1, and so there are no real solutions for $x$x and $y$y.
• Two lines: $4x^2-6y^2-24x+60y-114=0$4x26y224x+60y114=0 describes two intersecting lines, and not a hyperbola as expected.

Use the applet below to investigate the different curves represented by the general conic equation, by varying just the values of $A$A and $B$B.

• Can you find all four of the conic sections? Look at the relationship between $A$A and $B$B for each type.
• Can you find conditions under which the graph forms a line (or two lines)? Think about how this relates to planes intersecting a cone as in the image at the top.

Note: The values of $C$C, $D$D and $E$E are fixed in the applet at $-10$10, $10$10 and $10$10.

#### Worked examples

##### Example 1

The following equations describe different conic sections. Determine the conic section represented:

• $3x^2-2x+\frac{y^2}{4}+y-10=0$3x22x+y24+y10=0
• $-2x^2+7x+9y-2y^2+4=0$2x2+7x+9y2y2+4=0
• $x^2+10x-3y=0$x2+10x3y=0
• $5y^2+6x=5x^2-9y$5y2+6x=5x29y

Think: Since we already know that these equations represent conics, we can use the values of $A$A and $B$B to determine which conics they are. We can rearrange each equation so that it is in the form $Ax^2+By^2+Cx+Dy+E=0$Ax2+By2+Cx+Dy+E=0 and then read off the values.

Do:

• This equation doesn't need to be rearranged, $A=3$A=3 and $B=\frac{1}{4}$B=14. They are the same sign (both positive) but not equal, so the conic is an ellipse.
• The coefficients are $A=-2$A=2 and $B=-2$B=2. They are the same sign (both negative) and equal, so the conic is a circle.
• The coefficients are $A=1$A=1 and $B=0$B=0. $A$A or $B$B is zero, so the conic is a parabola.
• We need to rearrange this equation first to the form $5y^2+6x-5x^2+9y=0$5y2+6x5x2+9y=0. So $A=-5$A=5 and $B=5$B=5. The signs are opposite, so the conic is a hyperbola.

Reflect: We could only use this technique because the questions told us these equations definitely did describe conics. Otherwise we would have had to rearrange the equation to make sure they had a solution. Note also that if $A$A and $B$B have the same sign, it will be an ellipse, but if they are equal as well a circle is a more accurate description.

##### Example 2

Determine the curve produced by the equation $3x^2+5y^2-6x-20y+8=0$3x2+5y26x20y+8=0 and draw the graph.

Think: $A$A and $B$B are both positive so this looks like it will be an ellipse. We should first try to rearrange it into the form $\frac{\left(x-h\right)^2}{a^2}+\frac{\left(y-k\right)^2}{b^2}=1$(xh)2a2+(yk)2b2=1 to make sure it is an ellipse. We can then use the values of $a$a, $b$b, $h$h and $k$k to help draw the graph (if it is an ellipse).

Do: Rearrange the equation into the desired form by completing the square

 $3x^2+5y^2-6x-20y+8$3x2+5y2−6x−20y+8 $=$= $0$0 $3\left(x^2-2x\right)+5\left(y^2-4y\right)$3(x2−2x)+5(y2−4y) $=$= $-8$−8 $3\left(\left(x-1\right)^2-1\right)+5\left(\left(y-2\right)^2-4\right)$3((x−1)2−1)+5((y−2)2−4) $=$= $-8$−8 $3\left(x-1\right)^2+5\left(y-2\right)^2$3(x−1)2+5(y−2)2 $=$= $3+20-8$3+20−8 $\frac{3\left(x-1\right)^2}{15}+\frac{5\left(y-2\right)^2}{15}$3(x−1)215​+5(y−2)215​ $=$= $\frac{15}{15}$1515​ $\frac{\left(x-1\right)^2}{5}+\frac{\left(y-2\right)^2}{3}$(x−1)25​+(y−2)23​ $=$= $1$1

The equation describes a ellipse where $a=\sqrt{5}$a=5 and $b=\sqrt{3}$b=3, with a centre at $\left(1,2\right)$(1,2).

Notice the vertices at $\left(1\pm\sqrt{5},2\right)$(1±5,2) and co-vertices at $\left(1,2\pm\sqrt{3}\right)$(1,2±3).

##### Example 3

What does the following equation represent $2x^2-3y^2-12x+30y-57=0$2x23y212x+30y57=0? If it is a conic section, draw a graph of the conic.

Think: The coefficients of $x^2$x2 and $y^2$y2 have opposite signs, so if this is a conic section then it will be a hyperbola. We can try to rearrange the equation into the more useful hyperbola form $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$x2a2y2b2=1.

Do:

 $2x^2-3y^2-12x+30y-57$2x2−3y2−12x+30y−57 $=$= $0$0 $2\left(x^2-6x\right)-3\left(y^2-10\right)$2(x2−6x)−3(y2−10) $=$= $57$57 $2\left(\left(x-3\right)^2-9\right)-3\left(\left(y-5\right)^2-25\right)$2((x−3)2−9)−3((y−5)2−25) $=$= $57$57 $2\left(x-3\right)^2-18-3\left(y-5\right)^2+75$2(x−3)2−18−3(y−5)2+75 $=$= $57$57 $2\left(x-3\right)^2-3\left(y-5\right)^2$2(x−3)2−3(y−5)2 $=$= $57+18-75$57+18−75 $2\left(x-3\right)^2-3\left(y-5\right)^2$2(x−3)2−3(y−5)2 $=$= $0$0 $\left(y-5\right)^2$(y−5)2 $=$= $\frac{2}{3}\left(x-3\right)^2$23​(x−3)2 $y$y $=$= $\pm\sqrt{\frac{2}{3}}\left(x-3\right)+5$±√23​(x−3)+5

Reflect: The equation looks like it would be a hyperbola, however when we completed the squares there was no constant term. The result is just a pair of intersecting lines.

#### Practice questions

##### Question 1

Consider the conic section represented by the equation $x^2+y^2=4$x2+y2=4.

1. Which conic section does this best represent?

Hyperbola

A

Ellipse

B

Circle

C

Parabola

D

Hyperbola

A

Ellipse

B

Circle

C

Parabola

D
2. Draw the graph of the conic section.

##### Question 2

Consider the conic section represented by the equation $2x^2+8x-4y=0$2x2+8x4y=0.

1. Which conic section does this best represent?

Circle

A

Parabola

B

Hyperbola

C

Ellipse

D

Circle

A

Parabola

B

Hyperbola

C

Ellipse

D
2. Draw the graph of the conic section.

##### Question 3

Consider the conic section represented by the equation $25x^2+50x+4y^2-32y-11=0$25x2+50x+4y232y11=0.

1. Which conic section does this best represent?

Parabola

A

Ellipse

B

Circle

C

Hyperbola

D

Parabola

A

Ellipse

B

Circle

C

Hyperbola

D
2. Draw the graph of the conic section.

### Outcomes

#### M8-1

Apply the geometry of conic sections

#### 91573

Apply the geometry of conic sections in solving problems