FINDING THE VERTEX FOCUS AND DRECTRIX OF A PARABOLA FROM ITS EQUATION

In this section, you will learn how to find vertex, focus, equation of directrix and length of latus rectum of the parabola.

Before seeing example problems, let us remember some basic concepts about parabola. 

Parabola symmetric about x-axis and open right ward :

Standard form of parabola

y²  =  4ax

Parabola symmetric about x-axis and open left ward :

Standard form of parabola

y²  =  -4ax

Parabola symmetric about y-axis and open up ward :

Standard form of parabola

x²  =  4ay

Parabola symmetric about y-axis and open down ward :

Standard form of parabola

x²  =  -4ay

Now let us see some examples based on the above concept.

Example 1 :

Find the focus, vertex, equation of directrix and length of the latus rectum of the parabola

x²  =  5y

Solution :

From the given equation, the parabola is symmetric about y - axis and it is open upward.

x²  =  5y

4a  =  5

a  =  5/4

Vertex : V (0, 0)

Focus : F (0, 5/4)

Equation of directrix : y  =  -5/4

Length of latus rectum :  4a  =  4(5/4)  ==> 5

Example 2 :

Find the focus, vertex, equation of directrix and length of the latus rectum of the parabola

x² - 8y - 2x + 17  =  0

Solution :

x² - 8y - 2x + 17  =  0

x² - 2x  =  8y - 17

x² - 2x + 1² - 1²  =  8y - 17

(x - 1)²  =  8y - 17 + 1

(x - 1)²  =  8y - 16

(x - 1)²  =  8(y - 2)

From the given equation, the parabola is symmetric about y - axis and it is open upward.

Let X  =  x - 1 and Y  =  y - 2

X²  =  8Y

4a  =  8

a  =  2

Example 3 :

Find the focus, vertex, equation of directrix and length of the latus rectum of the parabola

 x²  =  -16y 

Solution :

From the given equation, the parabola is symmetric about y - axis and it is open downward.

x²  =  -16y

4a  =  16

a  =  4

Vertex : V (0, 0)

Focus : F (0, -4)

Equation of directrix : y  =  a  ==>  y  =  4

Length of latus rectum :  4a  =  4(4)  ==> 16

Example 4 :

Find the focus, vertex, equation of directrix and length of the latus rectum of the parabola

x² + 4y - 6x + 17  =  0

Solution :

x² + 4y - 6x + 17  =  0

x² - 6x  =  -4y - 17

x² - 6x + 3² - 3²  =  -4y - 17

(x - 3)²  =  -4y - 17 + 9

(x - 3)²  =  -4y - 8

(x - 3)²  =  -4(y + 2) 

From the given equation, the parabola is symmetric about y - axis and it is open downward.

Let X  =  x - 3 and Y  =  y + 2

X²  =  -4Y

4a  =  4

a  =  1

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