# Lesson Worksheet: Parabola Mathematics

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

**Q1: **

Find the focus and directrix of the parabola .

- Afocus: directrix:
- Bfocus: directrix:
- Cfocus: directrix:
- Dfocus: directrix:
- Efocus: directrix:

**Q2: **

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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**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 and whose directrix is the line .

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**Q5: **

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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**Q6: **

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

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**Q7: **

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

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**Q8: **

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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**Q9: **

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

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**Q10: **

The figure shows the parabola with its vertex marked.

What are the coordinates of ?

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