# 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.

**Q3: **

Find the equation of a parabola with a focus of (2, 2) and a directrix of . Give your answer in the form .

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**Q4: **

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

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**Q5: **

Consider the graph:

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

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**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?

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**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 . At what coordinates should you place the light bulb?

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**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 ,
- B ,
- C ,
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- E ,

**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 ft
- B ft
- C ft
- D ft
- E ft

**Q12: **

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 .

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

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Write an equation for the directrix.

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

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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.

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**Q13: **

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.

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**Q14: **

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

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**Q15: **

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

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**Q16: **

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.

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Write an expression for the distance between and the directrix .

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Equate the two expressions and square both sides.

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

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**Q17: **

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

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**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 -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 .

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**Q20: **

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).

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Write an expression for the distance between and the directrix .

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

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**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 .

Consider the parabola with Cartesian equation .

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

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Write the equation of its directrix.

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**Q22: **

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 .

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**Q23: **

Find the focus and directrix of the parabola .

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

**Q24: **

A parabola has the equation .

What are the coordinates of its focus?

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Write an equation for its directrix.

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**Q25: **

A parabola has the equation .

What are the coordinates of its focus?

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Write an equation for its directrix.

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