# HOW TO FIND VERTEX FOCUS AND DIRECTRIX OF A PARABOLA

## About "How to find vertex focus and directrix of a parabola"

How to find vertex focus and directrix of a parabola :

Before going to find these details first we have to check whether the equation of the parabola is in the standard form or not.

If it is in the standard then we can use the formula directly to find those details.

If the equation is not in the standard then we have to convert the given equation in standard form and use the formula.

## Parabola symmetric about x-axis and open right ward

Axis of symmetry : x -axis

Equation of axis : y = 0

Vertex V (0, 0)

Focus F (a, 0)

Equation of latus rectum : x = a

Equation of directrix : x = -a

length of latus rectum : 4a

Distance between directrix and latus rectum = 2a

how to find vertex focus and directrix of a parabola how to find vertex focus and directrix of a parabola

## Parabola symmetric about x-axis and open left ward

y² = -4ax is the standard equation of the parabola which is symmetric about x axis and open rightward.

Axis of symmetry : x -axis

Equation of axis : y = 0

Vertex V (0, 0)

Focus F (-a, 0)

Equation of latus rectum : x = -a

Equation of directrix : x = a

length of latus rectum : 4a

Distance between directrix and latus rectum = 2a

## Parabola symmetric about y-axis and open upward

x² = 4ay is the standard equation of the parabola which is symmetric about y axis and up rightward.

Axis of symmetry : y -axis

Equation of axis : x = 0

Vertex V (0, 0)

Focus F (0, a)

Equation of latus rectum : y = a

Equation of directrix : y = -a

length of latus rectum : 4a

Distance between directrix and latus rectum = 2a

## Parabola symmetric about y-axis and down upward

x² = -4ay is the standard equation of the parabola which is symmetric about y axis and down rightward.

Axis of symmetry : y -axis

Equation of axis : x = 0

Vertex V (0, 0)

Focus F (0, -a)

Equation of latus rectum : y = -a

Equation of directrix : y = a

length of latus rectum : 4a

Distance between directrix and latus rectum = 2a

Example 1 :

Find the axis, vertex, focus, directrix, equation of the latus rectum, length of latus rectum of the following parabola.

(y + 2)² = -8(x + 1)

Solution :

From the given information the parabola is symmetric about x -axis and leftward.

Let X = x + 1, Y = y + 2

(y + 2)² = -8(x + 1)

Y² = -8 X

4a = 8

a = 2

Example 2 :

Find the axis, vertex, focus, directrix, equation of the latus rectum, length of latus rectum of the following parabola.

x² - 2x + 8y + 17 = 0

Solution :

First we have to convert the given equation in standard form.

x² - 2x = - 8y - 17

x² - 2 x (1) + 1² - 1²  = - 8y - 17

(x - 1)² = -8y - 17 + 1

(x - 1)² = - 8y -16

(x - 1) = -8 (y + 2)

Let X = x - 1, Y = y + 2

X² = -8Y

4a = 8

a = 2

The given parabola is symmetric about X axis and open leftward.

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