FIND THE VERTEX OF THE PARABOLA

In the above equations (h, k) is the vertex of the parabola.

Let us look in to some example problems based on the above concept.

Example 1 :

Find the vertex of the following parabola

x² = - 4y

Solution :

x² = - 4y

We can compare the above equation with the general form (x - h)²  = -4 a (y - k)

(x - 0)² = - 4(y - 0)

So, the required vertex V(h, k) is (0, 0).

Example 2 :

Find the vertex of the following parabola

x² − 2x + 8y + 17 = 0

Solution :

x² − 2x + 8y + 17 = 0

Subtract 8y and 17 on both sides

x² − 2x + 8y + 17 - 8y - 17  =  -8y - 17

x² − 2x  =  -8y - 17

Split the coefficient of x as the multiple of 2.

x² − 2 ⋅x ⋅ 1 + 1² - 1²   =  -8y - 17

(x - 1)² - 1   =  -8y - 17

Add 1 on both sides

(x - 1)² - 1 + 1   =  -8y - 17 + 1

(x - 1)²   =  -8y - 16

(x - 1)²   =  -8(y + 2)

(x - 1)²   =  -8(y - (-2))

The above equation exactly matches with the equation

(x - h)²   =  -4a(y - k)

(h, k)  ==>  (1, -2)

So, the required vertex of the parabola is (1, -2).

Example 3 :

Find the vertex of the following parabola

y² − 8x + 6y + 9 = 0

Solution :

y² + 6y − 8x + 9  =  0

Add 8x and subtract 9 on both sides

y² + 6y − 8x + 9 + 8x - 9  =  0 + 8x - 9

y² + 6y  =  8x - 9

Split the coefficient of y as the multiple of 2.

y² + 2⋅ y ⋅ 3 + 3² - 3²  =  8x - 9

(y - 3)² - 9  =  8x - 9

Add 9 on both sides

(y - 3)² - 9 + 9  =  8x - 9 + 9

(y - 3)²  =  8x

(y - k)²  =  4a (x - h)

(y - (-3))²  =  8(x - 0)

(h, k)  ==>  (0, -3)

So, the required vertex of the parabola is (0, -3).

Example 4 :

Find the vertex of the following parabola

x² − 6x − 12y − 3 = 0

Solution :

x² − 6x − 12y − 3 = 0

Add 12y and 3 on both sides

x² − 6x − 12y − 3 + 12y + 3 = 0 + 12y + 3

x² − 6x  = 12y + 3

Split the coefficient of x as the multiple of 2.

x² − 2⋅x⋅3 + 3² - 3² = 12y + 3

(x - 3)² - 3² = 12y + 3

(x - 3)² - 9 = 12y + 3

Add 9 on both sides

(x - 3)² - 9 + 9 = 12y + 3 + 9

(x - 3)²  = 12y + 12

(x - 3)² = 12(y + 1)

(x - 3)² = 12(y - (-1))

(x - h)²  =  4a (y - k)

(x - 3)²  =  12(y - (-1))

(h, k)  ==>  (3, -1)

So, the required vertex of the parabola is (3, -1).

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