Worksheet: Identifying Conic Sections

In this worksheet, we will practice converting the general form of conic section equations into any of the standard forms.

Q1:

Consider the conic given by the equation 4𝑥+3𝑦32𝑥+6𝑦+55=0.

Write the equation in standard form.

  • A(𝑥4)+(𝑦+1)=12
  • B(𝑥4)3+(𝑦+1)4=1
  • C(𝑥+4)3(𝑦1)4=1
  • D(𝑥+4)3+(𝑦1)4=1
  • E(𝑥4)3(𝑦+1)4=1

Hence, describe the conic.

  • AAn ellipse with center (4,1)
  • BA circle with center (4,1)
  • CA hyperbola with center (4,1)
  • DA hyperbola with center (4,1)
  • EAn ellipse with center (4,1)

Q2:

Which type of conic is described by the equation 5𝑥9𝑦10𝑥+90𝑦265=0?

  • AA parabola
  • BA circle
  • CAn ellipse
  • DA hyperbola

Q3:

By calculating the discriminant, identify the type of conic that is described by the equation 𝑥+𝑦+10𝑥4𝑦+28=0.

  • AA parabola
  • BAn ellipse
  • CA circle
  • DA hyperbola

Q4:

The general equation of a conic has the form 𝐴𝑥+𝐵𝑥𝑦+𝐶𝑦+𝐷𝑥+𝐸𝑦+𝐹=0.

Consider the equation 2𝑥3𝑦16𝑥30𝑦49=0.

Calculate the value of the discriminant 𝐵4𝐴𝐶.

Hence, identify the conic described by the equation.

  • AParabola
  • BHyperbola
  • CEllipse
  • DCircle

Q5:

The general equation of a conic has the form 𝐴𝑥+𝐵𝑥𝑦+𝐶𝑦+𝐷𝑥+𝐸𝑦+𝐹=0.

Which of the following conditions would allow us to conclude that it is an ellipse?

  • A𝐵4𝐴𝐶<0 and either 𝐵0 or 𝐴𝐶
  • B𝐵4𝐴𝐶<0, 𝐵=0, and 𝐴=𝐶
  • C𝐵4𝐴𝐶>0
  • D𝐵4𝐴𝐶=0

Q6:

What are the possible values of 𝑚 that make the conic given by the equation 𝑥𝑦=𝑚 a hyperbola?

  • A𝜙
  • B
  • C{0}
  • D{1}

Q7:

Consider the conic given by the equation 𝑥+𝑘𝑥𝑦+𝑦+𝑥+𝑦+3=0. What is the set of all the possible values of 𝑘 that makes it a parabola?

  • A{4}
  • B{2}
  • C{2}
  • D{2,2}
  • E𝜙

Q8:

True or False: Any equation of a conic in which the 𝑥𝑦 term is missing is a circle equation.

  • ATrue
  • BFalse

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