Lesson: Equation of a Parabola

In this lesson, we will learn how to find the equation of a parabola using a focus point and a directrix equation or a vertex point and a directrix equation.

Sample Question Videos

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Worksheet: Equation of a Parabola • 25 Questions • 2 Videos

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 .

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 ?

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 .

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 .

Q5:

Consider the graph:

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

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?

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?

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.

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?

Q12:

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 .

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

Write an equation for the directrix.

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

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.

Q13:

What is the distance from the vertex to the directrix?

Find an equation for the parabola.

Q14:

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

Q15:

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

Q16:

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

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

Equate the two expressions and square both sides.

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

Q17:

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

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.

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 .

Q20:

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

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

Q21:

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

Write the equation of its directrix.

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 .

Q23:

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

Q24:

What are the coordinates of its focus?

Write an equation for its directrix.

Q25:

What are the coordinates of its focus?

Write an equation for its directrix.

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