# Worksheet: Equation of a Parabola

Q1:

The figure shows the parabola with its vertex marked.

What are the coordinates of ?

• A
• B
• C
• D
• E

Q2:

Find the equation of a parabola with a focus of and a directrix of . Give your answer in the form .

• A
• B
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• E

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.

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

Q4:

The diagram shows a parabola with a horizontal axis whose vertex is . The focus , directrix , and a point 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 .

Write the coordinates of the focus in terms of , , and .

• A
• B
• C
• D
• E

Write an expression for the distance from the point to the focus.

• A
• B
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• D
• E

Write an equation for the directrix.

• A
• B
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• D
• E

Write an expression for the distance between the point and the directrix.

• A
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• E

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 in terms of , , and that describes the parabola.

• A
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Q5:

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 ft
• B ft
• C ft
• D ft
• E ft

Q6:

A parabola has the equation .

What are the coordinates of its focus?

• A
• B
• C
• D
• E

Write an equation for its directrix.

• A
• B
• C
• D
• E

Q7:

Find an equation for the parabola whose focus is the point and whose directrix is the line .

• A
• B
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• E

Q8:

Find the equation of the parabola with focus and directrix . Give your answer in the form .

• A
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• E

Q9:

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 . At what coordinates should you place the light bulb?

• A
• B
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• D
• E

Q10:

The diagram shows a parabola that is symmetrical about the -axis and whose vertex is at the origin. Its Cartesian equation is , where is a positive constant. The focus of the parabola is the point and the directrix is the line with equation .

Find the Cartesian equation of the parabola whose focus is the point and whose directrix is the line .

• A
• B
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• D
• E

Q11:

Consider the parabola whose vertex is the point and whose directrix is the line .

What is the distance from the vertex to the directrix?

Find an equation for the parabola.

• A
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• E

Q12:

Consider the graph:

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

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Q13:

The given figure shows a parabola with a focus of , a directrix at , and a general point .

Find an expression for the length of the line from to the focus.

• A
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Write an expression for the distance between and the directrix .

• A
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Equate the two expressions and square both sides.

• A
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Expand and simplify the expressions excluding , and then make the subject and simplify.

• A
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Q14:

Find the focus and directrix of the parabola .

• A focus: directrix:
• B focus: directrix:
• C focus: directrix:
• D focus: directrix:
• E focus: directrix:

Q15:

A parabola has the equation .

What are the coordinates of its focus?

• A
• B
• C
• D
• E

Write an equation for its directrix.

• A
• B
• C
• D
• E

Q16:

A parabola has the equation .

What are the coordinates of its focus?

• A
• B
• C
• D
• E

Write an equation for its directrix.

• A
• B
• C
• D
• E

Q17:

A parabola has equation .

Find the coordinates of the vertex.

• A
• B
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• D
• E

Determine the equation of the directrix.

• A
• B
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• D
• E

Q18:

Write an equation for the parabola whose focus is the point and whose directrix is the line .

• A
• B
• C
• D
• E

Q19:

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?

Q20:

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?

Q21:

The diagram shows a parabola that is symmetrical about the -axis and whose vertex is at the origin. Its Cartesian equation is , where is a positive constant. The focus of the parabola is the point and the directrix is the line with equation .

Find the Cartesian equation of the parabola whose focus is the point and whose directrix is the line .

• A
• B
• C
• D
• E

Q22:

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

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

• A
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• D
• E

Write an expression for the distance between and the directrix .

• A
• B
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• D
• E

By equating the two expressions from and , work out an equation for the parabola. Give your answer in the form .

• A
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• E

Q23:

The diagram shows a parabola that is symmetrical about the -axis and whose vertex is at the origin. Its Cartesian equation is , where is a positive constant. The focus of the parabola is the point and the directrix is the line with equation .

Consider the parabola with Cartesian equation .

What are the coordinates of the focus of the parabola with Cartesian equation ?

• A
• B
• C
• D
• E

Write the equation of its directrix.

• A
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• C
• D
• E

Q24:

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 ,
• B ,
• C ,
• D ,
• E ,

Q25:

Write an equation for the parabola whose focus is the point and whose directrix is the line .

• A
• B
• C
• D
• E