Lesson Worksheet: Parabola Mathematics

Start Practising

In this worksheet, we will practice writing, solving, and graphing the equation of a parabola.

Q1:

Find the focus and directrix of the parabola 𝑦=2𝑥+5𝑥+4.

  • Afocus: 54,1, directrix: 𝑦=−43
  • Bfocus: −54,1, directrix: 𝑦=−43
  • Cfocus: −54,1, directrix: 𝑦=34
  • Dfocus: 44,1, directrix: 𝑦=34
  • Efocus: −45,1, directrix: 𝑦=34

Q2:

The diagram shows a parabola that is symmetrical about the 𝑦-axis and whose vertex is at the origin. Its Cartesian equation is 𝑥=4𝑝𝑦, where 𝑝 is a positive constant. The focus of the parabola is the point (0,𝑝) and the directrix is the line with equation 𝑦=−𝑝.

Find the Cartesian equation of the parabola whose focus is the point 0,54 and whose directrix is the line 𝑦=−54.

  • A𝑥=20𝑦
  • B𝑥=5𝑦
  • C𝑥=20𝑦
  • D𝑥=5𝑦
  • E𝑦=54𝑥

Q3:

Complete the following definition: A parabola is defined as the set of all points a fixed point called the focus and a fixed line called the directrix.

  • Awith a diameter from
  • Bequidistant from
  • Cat a given distance from
  • Dcentered between
  • Ewith a radius from

Q4:

Write an equation for the parabola whose focus is the point 0,−√23 and whose directrix is the line 𝑦=√23.

  • A𝑥=√23𝑦
  • B𝑦=4√23𝑥
  • C𝑦=√212𝑥
  • D𝑥=−4√23𝑦
  • E𝑥=√212𝑦

Q5:

The given figure shows a parabola with a focus of (𝑎,𝑏), a directrix at 𝑦=𝑘, and a general point (𝑥,𝑦).

Find an expression for the length of the line from (𝑥,𝑦) to the focus.

  • A√(𝑥−𝑎)+(𝑦−𝑏)
  • B√(𝑥−𝑎)−(𝑦−𝑏)
  • C√(𝑥+𝑎)+(𝑦+𝑏)
  • D√(𝑥+𝑎)+(𝑦+𝑏)
  • E√(𝑥−𝑎)+(𝑦−𝑏)

Write an expression for the distance between (𝑥,𝑦) and the directrix 𝑦=𝑘.

  • A𝑥+𝑘
  • B(𝑦−𝑘)
  • C𝑦−𝑘
  • D𝑥−𝑘
  • E𝑦+𝑘

Equate the two expressions and square both sides.

  • A(𝑥−𝑎)+(𝑦−𝑏)=(𝑦+𝑘)
  • B(𝑥−𝑏)+(𝑦−𝑎)=(𝑦−𝑘)
  • C(𝑥−𝑏)−(𝑦−𝑎)=(𝑦+𝑘)
  • D(𝑥−𝑎)−(𝑦−𝑏)=(𝑦−𝑘)
  • E(𝑥−𝑎)+(𝑦−𝑏)=(𝑦−𝑘)

Expand and simplify the expressions excluding (𝑥−𝑎), and then make 𝑦 the subject and simplify.

  • A𝑦=12(𝑥−𝑎)𝑏−𝑘+𝑏+𝑘
  • B𝑦=12(𝑥−𝑎)𝑏−𝑘−𝑏+𝑘
  • C𝑦=(𝑥−𝑎)𝑏+𝑘−𝑏+𝑘
  • D𝑦=12(𝑥−𝑎)𝑏−𝑘+𝑏−𝑘
  • E𝑦=12(𝑥−𝑎)𝑏+𝑘+𝑏−𝑘

Q6:

Find an equation for the parabola whose focus is the point (−5,−1) and whose directrix is the line 𝑦+12=0.

  • A(𝑥+5)=22(𝑦+1)
  • B(𝑥+5)=14(𝑦+1)
  • C(𝑥−5)=14(𝑦−1)
  • D(𝑥−5)=22(2𝑦−1)
  • E(𝑥+5)=12(𝑦+1)

Q7:

Write an equation for the parabola whose focus is the point (−4,−3) and whose directrix is the line 𝑥=0.

  • A(𝑦+3)=−2(𝑥+2)
  • B(𝑦−3)=−8(𝑥−2)
  • C(𝑦−3)=−8(𝑥−4)
  • D(𝑦+3)=−8(𝑥+4)
  • E(𝑦+3)=−8(𝑥+2)

Q8:

The diagram shows a parabola with a horizontal axis whose vertex is (ℎ,𝑘). The focus 𝐹, directrix 𝑑, and a point (𝑥,𝑦) on the parabola are marked.

The distance from the vertex to the focus is equal to the distance from the vertex to the directrix. Let this distance be 𝑝.

Write the coordinates of the focus in terms of ℎ, 𝑝, and 𝑘.

  • A(𝑘,ℎ−𝑝)
  • B(ℎ−𝑝,𝑘)
  • C(𝑘,ℎ+𝑝)
  • D(ℎ+𝑝,𝑘)
  • E(ℎ+𝑝,−𝑘)

Write an expression for the distance from the point (𝑥,𝑦) to the focus.

  • A(𝑥−(ℎ−𝑝))+(𝑦−𝑘)
  • B(𝑥−𝑘)+(𝑦−(ℎ−𝑝))
  • C(𝑥−𝑘)+(𝑦−(ℎ+𝑝))
  • D(𝑥−(ℎ+𝑝))+(𝑦−𝑘)
  • E(𝑥−(ℎ+𝑝))+(𝑦+𝑘)

Write an equation for the directrix.

  • A𝑥=ℎ𝑝
  • B𝑥=𝑝−ℎ
  • C𝑥=ℎ+𝑝
  • D𝑥=ℎ−𝑝
  • E𝑥=−ℎ𝑝

Write an expression for the distance between the point (𝑥,𝑦) and the directrix.

  • A𝑥−(ℎ𝑝)
  • B𝑥−(ℎ−𝑝)
  • C𝑥+(ℎ−𝑝)
  • D𝑥+(ℎ+𝑝)
  • E𝑥−(ℎ+𝑝)

A parabola can be defined as the locus of points that are equidistant from a fixed line (the directrix) and a fixed point that is not on the line (the focus).

By equating your expressions, squaring both sides, and rearranging, write an equation for (𝑦−𝑘) in terms of 𝑥, 𝑝, and ℎ that describes the parabola.

  • A(𝑦+𝑘)=𝑝(𝑥+ℎ)
  • B(𝑦+𝑘)=4𝑝(𝑥+ℎ)
  • C(𝑦−ℎ)=4𝑝(𝑥−𝑘)
  • D(𝑦−𝑘)=𝑝(𝑥−ℎ)
  • E(𝑦−𝑘)=4𝑝(𝑥−ℎ)

Q9:

Find the equation of the parabola with focus (3,−2) and directrix 𝑦=−32. Give your answer in the form 𝑦=𝑎𝑥+𝑏𝑥+𝑐.

  • A𝑦=−𝑥+6𝑥−212
  • B𝑦=𝑥+6𝑥−212
  • C𝑦=−𝑥+6𝑥−433
  • D𝑦=𝑥+6𝑥+434
  • E𝑦=−𝑥+6𝑥+434

Q10:

The figure shows the parabola 𝑥=2𝑦−16𝑦+22 with its vertex 𝑉 marked.

What are the coordinates of 𝑉?

  • A(−6,4)
  • B(−4,10)
  • C(−4,6)
  • D(4,−10)
  • E(−10,4)

Practice Means Progress

Boost your grades with free daily practice questions. Download Nagwa Practice today!

scan me!

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.