Worksheet: Equation of a Parabola

In this worksheet, we will practice finding the equation of a parabola using a focus point and a directrix equation or a vertex point and a directrix equation.

Q1:

Find the equation of a parabola with a focus of ( 1 , 3 ) and a directrix of 𝑦 = 5 . Give your answer in the form 𝑦 = 𝑎 𝑥 + 𝑏 𝑥 + 𝑐 2 .

  • A 𝑦 = 1 2 𝑥 1 4 𝑥 1 5 4 2
  • B 𝑦 = 1 2 𝑥 + 1 4 𝑥 + 1 5 4 2
  • C 𝑦 = 1 4 𝑥 + 1 2 𝑥 + 1 5 4 2
  • D 𝑦 = 1 4 𝑥 + 1 2 𝑥 1 5 4 2
  • E 𝑦 = 1 2 𝑥 + 𝑥 1 5 4 2

Q2:

The figure shows the parabola 𝑥 = 2 𝑦 1 6 𝑦 + 2 2 2 with its vertex 𝑉 marked.

What are the coordinates of 𝑉 ?

  • A ( 4 , 1 0 )
  • B ( 6 , 4 )
  • C ( 4 , 6 )
  • D ( 1 0 , 4 )
  • E ( 4 , 1 0 )

Q3:

Find the equation of a parabola with a focus of (2, 2) and a directrix of 𝑦 = 1 . Give your answer in the form 𝑦 = 𝑎 𝑥 + 𝑏 𝑥 + 𝑐 2 .

  • A 𝑦 = 1 6 𝑥 3 2 𝑥 + 1 6 2
  • B 𝑦 = 1 6 𝑥 + 2 3 𝑥 + 7 6 2
  • C 𝑦 = 1 6 𝑥 3 2 𝑥 + 1 6 2
  • D 𝑦 = 1 6 𝑥 2 3 𝑥 + 7 6 2
  • E 𝑦 = 1 3 𝑥 4 3 𝑥 + 7 6 2

Q4:

Find the equation of the parabola with focus ( 3 , 2 ) and directrix 𝑦 = 3 2 . Give your answer in the form 𝑦 = 𝑎 𝑥 + 𝑏 𝑥 + 𝑐 2 .

  • A 𝑦 = 𝑥 + 6 𝑥 + 4 3 4 2
  • B 𝑦 = 𝑥 + 6 𝑥 + 4 3 4 2
  • C 𝑦 = 𝑥 + 6 𝑥 2 1 2 2
  • D 𝑦 = 𝑥 + 6 𝑥 4 3 3 2
  • E 𝑦 = 𝑥 + 6 𝑥 2 1 2 2

Q5:

Consider the graph:

Which of the following could be the equation of the parabola?

  • A 𝑦 = ( 𝑥 1 ) ( 𝑥 5 )
  • B 𝑦 = ( 𝑥 + 1 ) ( 𝑥 + 5 )
  • C 𝑦 = ( 𝑥 1 ) ( 𝑥 5 )
  • D 𝑦 = ( 𝑥 + 1 ) ( 𝑥 + 5 )
  • E 𝑦 = ( 𝑥 + 1 ) ( 𝑥 5 )

Q6:

If we want to construct a mirror in an automobile headlight having a parabolic cross section with the light bulb at the focus, and the focus is located at (0, 0.25), what should the equation of the parabola be?

  • A 𝑦 = 𝑥 2
  • B 𝑥 = 0 . 2 5 𝑦 2
  • C 𝑦 = 0 . 2 5 𝑥 2
  • D 𝑥 = 𝑦 2
  • E 𝑦 = 4 𝑥 2

Q7:

The mirror in an automobile headlight has a parabolic cross section with the light bulb at the focus. On a schematic, the equation of the parabola is given as 𝑥 = 4 𝑦 2 . At what coordinates should you place the light bulb?

  • A ( 1 , 0 )
  • B ( 0 , 1 )
  • C ( 1 , 0 )
  • D ( 0 , 1 )
  • E ( 2 , 1 )

Q8:

A satellite dish is shaped like a paraboloid of revolution. This means that it can be formed by rotating a parabola around its axis of symmetry. The receiver is to be located at the focus. If the dish is 12 feet across at its opening and 4 feet deep at its center, how far should the receiver be placed above the vertex?

Q9:

A satellite dish is shaped like a paraboloid of revolution. This means that it can be formed by rotating a parabola around its axis of symmetry. The receiver is to be located at the focus. If the dish is 8 feet across at its opening and 2 feet deep, how far should the receiver be placed above the vertex?

Q10:

An arch is in the shape of a parabola. It has a span of 100 feet and a maximum height of 20 feet. Find the equation of the parabola, and determine the height of the arch 40 feet from the center.

  • A 𝑥 = 4 ( 3 1 . 2 5 𝑦 2 0 ) 2 , = 1 3 . 4 4 f t
  • B 𝑥 = 1 2 5 𝑦 2 , = 1 2 . 8 f t
  • C 𝑥 = 1 2 5 𝑦 2 0 2 , = 1 2 . 9 6 f t
  • D 𝑥 = 1 2 5 ( 𝑦 2 0 ) 2 , = 7 . 2 f t
  • E 𝑥 = 1 2 5 𝑦 2 0 2 , = 7 . 2 f t

Q11:

An arch in the shape of a parabola has a span of 160 feet and a maximum height of 40 feet. Find the equation of the parabola. At what distance from the center is the arch 20 ft high?

  • A 𝑥 = 1 6 0 ( 𝑦 + 4 0 ) , 𝑥 = 4 0 6 2 ft
  • B 𝑥 = 1 6 0 𝑦 , 𝑥 = 4 0 2 2 ft
  • C 𝑦 = 1 6 0 ( 𝑥 4 0 ) , 𝑥 = 3 7 . 5 2 ft
  • D 𝑥 = 1 6 0 ( 𝑦 4 0 ) , 𝑥 = 4 0 2 2 ft
  • E 𝑥 = 4 ( 4 0 𝑦 4 0 ) , 𝑥 = 4 2 1 0 2 ft

Q12:

The diagram shows a parabola with a horizontal axis whose vertex is ( , 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 , 𝑝 , and 𝑘 .

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

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

  • A ( 𝑥 ( + 𝑝 ) ) + ( 𝑦 𝑘 ) 2 2
  • B ( 𝑥 𝑘 ) + ( 𝑦 ( 𝑝 ) ) 2 2
  • C ( 𝑥 𝑘 ) + ( 𝑦 ( + 𝑝 ) ) 2 2
  • D ( 𝑥 ( 𝑝 ) ) + ( 𝑦 𝑘 ) 2 2
  • E ( 𝑥 ( + 𝑝 ) ) + ( 𝑦 + 𝑘 ) 2 2

Write an equation for the directrix.

  • A 𝑥 = 𝑝
  • B 𝑥 = + 𝑝
  • C 𝑥 = 𝑝
  • D 𝑥 = 𝑝
  • E 𝑥 = 𝑝

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

  • A 𝑥 ( 𝑝 )
  • B 𝑥 + ( + 𝑝 )
  • C 𝑥 + ( 𝑝 )
  • D 𝑥 ( + 𝑝 )
  • E 𝑥 ( 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 ) 2 in terms of 𝑥 , 𝑝 , and that describes the parabola.

  • A ( 𝑦 + 𝑘 ) = 𝑝 ( 𝑥 + ) 2
  • B ( 𝑦 + 𝑘 ) = 4 𝑝 ( 𝑥 + ) 2
  • C ( 𝑦 𝑘 ) = 𝑝 ( 𝑥 ) 2
  • D ( 𝑦 𝑘 ) = 4 𝑝 ( 𝑥 ) 2
  • E ( 𝑦 ) = 4 𝑝 ( 𝑥 𝑘 ) 2

Q13:

Consider the parabola whose vertex is the point ( 5 , 4 ) and whose directrix is the line 𝑥 = 1 .

What is the distance from the vertex to the directrix?

Find an equation for the parabola.

  • A ( 𝑦 4 ) = 1 6 ( 𝑥 + 5 ) 2
  • B ( 𝑦 4 ) = 2 0 ( 𝑥 + 5 ) 2
  • C ( 𝑦 + 4 ) = 1 6 ( 𝑥 5 ) 2
  • D ( 𝑦 4 ) = 4 ( 𝑥 + 5 ) 2
  • E ( 𝑦 + 4 ) = 2 0 ( 𝑥 5 ) 2

Q14:

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

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

Q15:

Find an equation for the parabola whose focus is the point ( 5 , 1 ) and whose directrix is the line 𝑦 + 1 2 = 0 .

  • A ( 𝑥 + 5 ) = 1 4 ( 𝑦 + 1 ) 2 2
  • B ( 𝑥 5 ) = 2 2 ( 2 𝑦 1 ) 2 2
  • C ( 𝑥 5 ) = 1 4 ( 𝑦 1 ) 2 2
  • D ( 𝑥 + 5 ) = 2 2 ( 𝑦 + 1 ) 2 2
  • E ( 𝑥 + 5 ) = 1 2 ( 𝑦 + 1 ) 2 2

Q16:

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 ) 2 2
  • B ( 𝑥 𝑎 ) ( 𝑦 𝑏 ) 2 2
  • C ( 𝑥 𝑎 ) + ( 𝑦 𝑏 )
  • D ( 𝑥 𝑎 ) + ( 𝑦 𝑏 ) 2 2
  • E ( 𝑥 + 𝑎 ) + ( 𝑦 + 𝑏 )

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

  • A 𝑦 𝑘
  • B 𝑥 + 𝑘
  • C 𝑥 𝑘
  • D 𝑦 + 𝑘
  • E ( 𝑦 𝑘 ) 2

Equate the two expressions and square both sides.

  • A ( 𝑥 𝑏 ) ( 𝑦 𝑎 ) = ( 𝑦 + 𝑘 ) 2 2 2
  • B ( 𝑥 𝑎 ) + ( 𝑦 𝑏 ) = ( 𝑦 + 𝑘 ) 2 2 2
  • C ( 𝑥 𝑎 ) + ( 𝑦 𝑏 ) = ( 𝑦 𝑘 ) 2 2 2
  • D ( 𝑥 𝑏 ) + ( 𝑦 𝑎 ) = ( 𝑦 𝑘 ) 2 2 2
  • E ( 𝑥 𝑎 ) ( 𝑦 𝑏 ) = ( 𝑦 𝑘 ) 2 2 2

Expand and simplify the expressions excluding ( 𝑥 𝑎 ) 2 , and then make 𝑦 the subject and simplify.

  • A 𝑦 = 1 2 ( 𝑥 𝑎 ) 𝑏 𝑘 + 𝑏 + 𝑘 2
  • B 𝑦 = ( 𝑥 𝑎 ) 𝑏 + 𝑘 𝑏 + 𝑘 2
  • C 𝑦 = 1 2 ( 𝑥 𝑎 ) 𝑏 𝑘 𝑏 + 𝑘 2
  • D 𝑦 = 1 2 ( 𝑥 𝑎 ) 𝑏 𝑘 + 𝑏 𝑘 2
  • E 𝑦 = 1 2 ( 𝑥 𝑎 ) 𝑏 + 𝑘 + 𝑏 𝑘 2

Q17:

Write an equation for the parabola whose focus is the point 0 , 2 3 and whose directrix is the line 𝑦 = 2 3 .

  • A 𝑥 = 2 1 2 𝑦 2
  • B 𝑥 = 2 3 𝑦 2
  • C 𝑦 = 4 2 3 𝑥 2
  • D 𝑥 = 4 2 3 𝑦 2
  • E 𝑦 = 2 1 2 𝑥 2

Q18:

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.

  • Acentered between
  • Bat a given distance from
  • Cwith a radius from
  • Dequidistant from
  • Ewith a diameter from

Q19:

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

Find the Cartesian equation of the parabola whose focus is the point 3 2 , 0 and whose directrix is the line 𝑥 + 3 2 = 0 .

  • A 𝑦 = 6 𝑥
  • B 𝑦 = 3 2 𝑥
  • C 𝑦 = 1 2 𝑥
  • D 𝑦 = 6 𝑥
  • E 𝑦 = 1 2 𝑥

Q20:

The given figure shows a parabola with a focus of (3, 2), a directrix at 𝑦 = 1 , and a general point ( 𝑥 , 𝑦 ) .

Find an expression for the length of the line from the point ( 𝑥 , 𝑦 ) to the point (3, 2).

  • A ( 𝑥 2 ) + ( 𝑦 3 )
  • B ( 𝑥 3 ) ( 𝑦 2 )
  • C ( 𝑥 2 ) ( 𝑦 3 )
  • D ( 𝑥 3 ) + ( 𝑦 2 )
  • E ( 𝑥 3 ) + ( 𝑦 2 )

Write an expression for the distance between ( 𝑥 , 𝑦 ) and the directrix 𝑦 = 1 .

  • A 𝑦 1
  • B 𝑥 + ( 𝑦 1 )
  • C 𝑥 1
  • D 𝑥 ( 𝑦 1 )
  • E ( 𝑦 1 )

By equating the two expressions from ( 𝑎 ) and ( 𝑏 ) , work out an equation for the parabola. Give your answer in the form 𝑦 = 𝑎 𝑥 + 𝑏 𝑥 + 𝑐 .

  • A 𝑦 = 1 6 𝑥 𝑥 + 2
  • B 𝑦 = 1 2 𝑥 3 𝑥 + 7
  • C 𝑦 = 1 2 𝑥 3 𝑥 + 6
  • D 𝑦 = 1 2 𝑥 + 3 𝑥 + 6
  • E 𝑦 = 1 2 𝑥 + 3 𝑥 + 7

Q21:

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

Consider the parabola with Cartesian equation 𝑦 = 1 4 𝑥 .

What are the coordinates of the focus of the parabola with Cartesian equation 𝑦 = 1 4 𝑥 ?

  • A 7 2 , 0
  • B 0 , 7 2
  • C 0 , 7 2
  • D 7 2 , 0
  • E ( 1 4 , 0 )

Write the equation of its directrix.

  • A 𝑥 = 7 2
  • B 𝑥 = 1 4
  • C 𝑥 = 1 4
  • D 𝑥 = 7 2
  • E 𝑥 = 7

Q22:

The diagram shows a parabola that is symmetrical about the 𝑦 -axis and whose vertex is at the origin. Its Cartesian equation is 𝑥 = 4 𝑝 𝑦 , 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 , 5 4 and whose directrix is the line 𝑦 = 5 4 .

  • A 𝑥 = 5 𝑦
  • B 𝑦 = 5 4 𝑥
  • C 𝑥 = 2 0 𝑦
  • D 𝑥 = 5 𝑦
  • E 𝑥 = 2 0 𝑦

Q23:

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

  • A focus: 4 5 , 1 , directrix: 𝑦 = 3 4
  • B focus: 5 4 , 1 , directrix: 𝑦 = 4 3
  • C focus: 4 4 , 1 , directrix: 𝑦 = 3 4
  • D focus: 5 4 , 1 , directrix: 𝑦 = 3 4
  • E focus: 5 4 , 1 , directrix: 𝑦 = 4 3

Q24:

A parabola has the equation 𝑥 = 3 2 𝑦 2 .

What are the coordinates of its focus?

  • A ( 0 , 6 )
  • B 3 8 , 0
  • C ( 6 , 0 )
  • D 0 , 3 8
  • E 0 , 3 2

Write an equation for its directrix.

  • A 𝑦 + 3 8 = 0
  • B 𝑦 6 = 0
  • C 𝑦 + 6 = 0
  • D 𝑦 3 8 = 0
  • E 𝑦 + 3 2 = 0

Q25:

A parabola has the equation 𝑥 = 2 2 𝑦 2 .

What are the coordinates of its focus?

  • A 0 , 8 2
  • B 2 2 , 0
  • C 8 2 , 0
  • D 0 , 2 2
  • E 0 , 2 2

Write an equation for its directrix.

  • A 𝑦 + 2 2 = 0
  • B 𝑦 8 2 = 0
  • C 𝑦 + 8 2 = 0
  • D 𝑦 2 2 = 0
  • E 𝑦 + 2 2 = 0

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