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# Classifying Conic Sections

## Use the discriminant to classify ellipses, parabolas, and hyperbolas

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Practice Classifying Conic Sections
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Classifying Conic Sections

You and your friends are playing Name the Conic Section. Your friend pulls a card with the equation $x^2 + 3xy = -5y^2 -10$ written on it. What type of conic section is represented by the equation?

### Guidance

Another way to classify a conic section when it is in the general form is to use the discriminant, like from the Quadratic Formula. The discriminant is what is underneath the radical, $b^2-4ac$ , and we can use this to determine if the conic is a parabola, circle, ellipse, or hyperbola. If the general form of the equation is $Ax^2+Bxy+Cy^2+Dx+Ey+F=0$ , where $B=0$ , then the discriminant will be $B^2-4AC$ .

Use the table below:

$B^2-4AC=0$ and $A=0$ or $C=0$ Parabola
$B^2-4AC<0$ and $A=C$ Circle
$B^2-4AC<0$ and $A \ne C$ Ellipse
$B^2-4AC>0$ Hyperbola

#### Example A

Use the discriminant to determine the type conic: $x^2-4y^2+5x-8y+16=0$ .

Solution: $A=1$ , $B=0$ , and $C=-4$

$0^2-4(1)(-4)=16$ This is a hyperbola.

#### Example B

Use the discriminant to determine the type of conic: $3x^2+3y^2-9x-12y-20=0$

Solution: $A=3$ , $B=0$ , $C=3$

$0^2-4(3)(3)=-36$ Because $A=C$ and the discriminant is less than zero, this is a circle.

#### Example C

Use the discriminant to determine the type of conic. Then, change the equation into standard form to verify your answer. Find the center or vertex, if it is a parabola. $x^2+y^2-6x+14y-86=0$

Solution: $A=1, \ B=0, \ C=1$ This is a circle.

$(x^2-6x+9)+(y^2+14y+49)&=86+49+9 \\(x-3)^2+(y+7)^2 &=144$

The center is $(3, \ -7)$ .

Intro Problem Revisit First we need to rewrite the equation is standard form.

$x^2 + 3xy = -5y^2 -10x^2 + 3xy + 5y^2 + 10 = 0$

Now we can use the discriminant to find the type of conic section represented by the equation.

$A=1, \ B=3, \ C=5$

$3^2-4(1)(5)=-11$ Because $A \ne C$ and the discriminant is less than zero, this equation represents an ellipse.

### Guided Practice

Use the discriminant to determine the type of conic.

1. $2x^2+5y^2-8x+25y+115=0$

2. $5y^2-9x-10y-14=0$

3. Use the discriminant to determine the type of conic. Then, change the equation into standard form to verify your answer. Find the center or vertex, if it is a parabola.

$-4x^2+3y^2-8x+24y+32=0$

1. $0^2-4(2)(5)=-40$ , this is an ellipse.

2. $0^2-4(0)(5)=0$ , this is a parabola.

3. $0^2-4(-4)(3)=48$ , this is a hyperbola. Changing it to standard form, we have:

$(-4x^2-8x)+(3y^2+24y) &=-32 \\-4(x^2+2x+1)+3(y^2+8y+16) &=-32+48-4 \\-4(x+1)^2+3(y+4)^2 &=12 \\\frac{-(x+1)^2}{3}+\frac{(y+4)^2}{4} &=1$

Usually, we write the negative term second, so the equation is $\frac{(y+4)^2}{4} - \frac{(x+1)^2}{3}=1$ . The center is $(-1, -4)$ .

### Vocabulary

Discriminant
When referring to the general second-degree equation, $Ax^2+Bxy+Cy^2+Dx+Ey+F=0$ , the discriminant is $B^2-4AC$ and it determines the type of conic the equation represents.

### Practice

Use the discriminant to determine the type of conic each equation represents.

1. $2x^2+2y^2+16x-8y+25=0$
2. $x^2-y^2-2x+5y-12=0$
3. $6x^2+y^2-12x+7y+35=0$
4. $3x^2-15x+9y-18=0$
5. $10y^2+6x-40y+253=0$
6. $4x^2+4y^2+32x+48y+465=0$

Match the equation with the correct graph.

1. $x^2+10x+4y+41=0$
2. $4y^2+x+56y+188=0$
3. $x^2+y^2+10x-14y+65=0$
4. $25x^2+y^2-200x-10y+400=0$

Use the discriminant to determine the type of conic. Then, change the equation into standard form to verify your answer. Find the center or vertex, if it is a parabola.

1. $x^2-12x+6y+66=0$
2. $x^2+y^2+2x+2y-2=0$
3. $x^2-y^2-10x-10y-10=0$
4. $y^2-10x+8y+46=0$
5. Find the Area of an Ellipse Graph $x^2+y^2=36$ and find is area.
1. Then, graph $\frac{x^2}{36} + \frac{y^2}{25}=1$ and $\frac{x^2}{25} + \frac{y^2}{36}=1$ on the same axes.
2. Do these ellipses have the same area? Why or why not?
3. If the equation of the area of a circle is $A=\pi r^2$ , what do you think the area of an ellipse is? Use $a$ and $b$ as in the standard form, $\frac{x^2}{a^2} + \frac{y^2}{b^2}=1$ .
4. Find the areas of the ellipses from part a. Are the areas more or less than the area of the circle? Why or why not?