Find the equation of a parabola with a focus at (–2,–3) and a directrix at y = –5.

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Popular Problems Algebra Find the Parabola with Focus (2,-3) and Directrix x=5 (2,-3) x=5

(

2

,

−

3

)

x

=

5

Since the directrix is horizontal, use the equation of a parabola that opens left or right.

(

y

−

k

)

2

=

4

p

(

x

−

h

)

Find the vertex.

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The vertex

(

h

,

k

)

is halfway between the directrix and focus. Find the

x

coordinate of the vertex using the formula

x

=

x coordinate of focus

+

directrix

2

. The

y

coordinate will be the same as the

y

coordinate of the focus.

(

2

+

5

2

,

−

3

)

Add

2

and

5

.

(

7

2

,

−

3

)

Find the distance from the focus to the vertex.

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The distance from the focus to the vertex and from the vertex to the directrix is

|

p

|

. Subtract the

x

coordinate of the vertex from the

x

coordinate of the focus to find

p

.

p

=

2

−

7

2

Simplify.

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p

=

−

3

2

Substitute in the known values for the variables into the equation

(

y

−

k

)

2

=

4

p

(

x

−

h

)

.

(

y

+

3

)

2

=

4

(

−

3

2

)

(

x

−

7

2

)

Simplify.

(

y

+

3

)

2

=

−

6

(

x

−

7

2

)

50-60=10

step-by-step explanation: