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Classifying Conic Sections

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Classifying Conic Sections
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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

Answers

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?

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