Find an equation of the parabola with vertex and directrix

Given the focus and directrix of a parabola , how do we find the equation of the parabola?

If we consider only parabolas that open upwards or downwards, then the directrix will be a horizontal line of the form y = c .

Let ( a , b ) be the focus and let y = c be the directrix. Let ( x 0 , y 0 ) be any point on the parabola.

Any point, ( x 0 , y 0 ) on the parabola satisfies the definition of parabola, so there are two distances to calculate:

1. Distance between the point on the parabola to the focus
2. Distance between the point on the parabola to the directrix

To find the equation of the parabola, equate these two expressions and solve for y 0 .

Find the equation of the parabola in the example above.

Distance between the point ( x 0 , y 0 ) and ( a , b ) :

( x 0 − a ) 2 + ( y 0 − b ) 2

Distance between point ( x 0 , y 0 ) and the line y = c :

| y 0 − c |

(Here, the distance between the point and horizontal line is difference of their y -coordinates.)

Equate the two expressions.

( x 0 − a ) 2 + ( y 0 − b ) 2 = | y 0 − c |

Square both sides.

( x 0 − a ) 2 + ( y 0 − b ) 2 = ( y 0 − c ) 2

Expand the expression in y 0 on both sides and simplify.

( x 0 − a ) 2 + b 2 − c 2 = 2 ( b − c ) y 0

This equation in ( x 0 , y 0 ) is true for all other values on the parabola and hence we can rewrite with ( x , y ) .

Therefore, the equation of the parabola with focus ( a , b ) and directrix y = c is

( x − a ) 2 + b 2 − c 2 = 2 ( b − c ) y

Example:

If the focus of a parabola is ( 2 , 5 ) and the directrix is y = 3 , find the equation of the parabola.

Let ( x 0 , y 0 ) be any point on the parabola. Find the distance between ( x 0 , y 0 ) and the focus. Then find the distance between ( x 0 , y 0 ) and directrix. Equate these two distance equations and the simplified equation in x 0 and y 0 is equation of the parabola.

The distance between ( x 0 , y 0 ) and ( 2 , 5 ) is ( x 0 − 2 ) 2 + ( y 0 − 5 ) 2

The distance between ( x 0 , y 0 ) and the directrix, y = 3 is

| y 0 − 3 | .

Equate the two distance expressions and square on both sides.

( x 0 − 2 ) 2 + ( y 0 − 5 ) 2 = | y 0 − 3 |

( x 0 − 2 ) 2 + ( y 0 − 5 ) 2 = ( y 0 − 3 ) 2

Simplify and bring all terms to one side:

x 0 2 − 4 x 0 − 4 y 0 + 20 = 0

Write the equation with y 0 on one side:

y 0 = x 0 2 4 − x 0 + 5

This equation in ( x 0 , y 0 ) is true for all other values on the parabola and hence we can rewrite with ( x , y ) .

So, the equation of the parabola with focus ( 2 , 5 ) and directrix is y = 3 is

y = x 2 4 − x + 5

Created by Bogna Szyk and Wojciech Sas, PhD candidate

Reviewed by Steven Wooding and Jack Bowater

Last updated: Jan 18, 2022

Any time you come across a quadratic formula you want to analyze, you'll find this parabola calculator to be the perfect tool for you. Not only will it provide you with the parabola equation in both the standard form and the vertex form, but also calculate the parabola vertex, focus, and directrix for you.

What is a parabola?

source: Wikimedia

A parabola is a U-shaped symmetrical curve. Its main property is that every point lying on the parabola is equidistant from both a certain point, called the focus of a parabola, and a line, called its directrix. It is also the curve that corresponds to quadratic equations.

The axis of symmetry of a parabola is always perpendicular to the directrix and goes through the focus point. The vertex of a parabola is the point at which the parabola makes its sharpest turn; it lies halfway between the focus and the directrix.

A real-life example of a parabola is the path traced by an object in projectile motion.

The parabola equation in vertex form

The standard form of a quadratic equation is y = ax² + bx + c. You can use this vertex calculator to transform that equation into the vertex form, which allows you to find the important points of the parabola – its vertex and focus.

The parabola equation in its vertex form is y = a(x - h)² + k, where:

• a — Same as the a coefficient in the standard form;
• h — x-coordinate of the parabola vertex; and
• k — y-coordinate of the parabola vertex.

You can calculate the values of h and k from the equations below:

Parabola focus and directrix

The parabola vertex form calculator also finds the focus and directrix of the parabola. All you have to do is to use the following equations:

• Focus x-coordinate: x₀ = - b/(2a);
• Focus y-coordinate: y₀ = c - (b² - 1)/(4a); and
• Directrix equation: y = c - (b² + 1)/(4a).

How to use the parabola equation calculator: an example

1. Enter the coefficients a, b and c of the standard form of your quadratic equation. Let's assume that the equation is y = 2x² + 3x - 4, what means that a = 2, b = 3 and c = -4.

2. Calculate the coordinates of the vertex, using the formulas listed above:

   h = - b/(2a) = -3/4 = -0.75

   k = c - b²/(4a) = -4 - 9/8 = -5.125

3. Find the coordinates of the focus of the parabola. The x-coordinate of the focus is the same as the vertex's (x₀ = -0.75), and the y-coordinate is:

   y₀ = c - (b² - 1)/(4a) = -4 - (9-1)/8 = -5

4. Find the directrix of the parabola. You can either use the parabola calculator to do it for you, or you can use the equation:

   y = c - (b² + 1)/(4a) = -4 - (9+1)/8 = -5.25

If you want to learn more coordinate geometry concepts, we recommend checking the average rate of change calculator and the latus rectum calculator.

FAQ

What is a parabola?

A parabola is a symmetrical U shaped curve such that every point on the curve is equidistant from the directrix and the focus.

How do I define a parabola?

A parabola is defined by the equation such that every point on the curve satisfies it. Mathematically, y = ax² + bx + c.

How do I calculate the vertex of a parabola?

To calculate the vertex of a parabola defined by coordinates (x, y):

1. Find x coordinate using the axis of symmetry formula:

   x₀ = - b/(2a)

2. Find y coordinate using the equation of parabola:

   y₀ = c - (b² - 1)/(4a)

How to calculate the focus of a parabola?

To calculate the focus of a parabola defined by coordinates (x, y):

1. Find y coordinate using the formula y = c - (b² + 1)/(4a)
2. Find x coordinate using the parabola equation.

Bogna Szyk and Wojciech Sas, PhD candidate

What to input?

Standard form: y = ax² + bx + c

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What is the equation of the parabola with focus and Directrix?