Find Coordinates of Focus of Parabola Method 1

The following steps can be performed to get the coordinates of the focus of a parabola from its equation:

1. Convert the equation to vertex form `y = a(x - h)^2 + k`
2. Identify ‘a’, ‘h’ and ‘k’
3. Focus of the parabola is `(h, k + 1/(4a))`

The following examples will make it clearer.

Example 1

Find the focus of a parabola with the equation `y = 3(x – 1)^2 + 2`

Solution

Step 1: Convert the equation to vertex form `y = a(x - h)^2 + k`

The given equation is already in the vertex form. Thus we don’t need to convert it into the vertex form.

Step 2: Identify ‘a’, ‘h’ and ‘k’

Comparing the equation `y = 3(x – 1)^2  + 2` with the vertex form `y = a(x – h)^2 + k` we get

`a = 3`
`h = 1`
`k = 2`

Step3: Focus of the parabola is `(h, k + 1/(4a))`

Plug in the values of ‘a’, ‘h’ and ‘k’ into the above formula to get the coordinates of focus.

`(1, 2 + 1/(4*3))`

Simplify,

`(1, 25/12)`

Thus, the coordinates of the focus of the parabola `y = 3(x – 1)^2 + 2` are `(1, 25/12)`.

Example 2

Find the focus of a parabola with the equation `y = 2x^2 + 3x + 1`

Solution

Step 1: Convert the equation to vertex form `y = a(x - h)^2 + k`

We will do this by using the completing the square method. First, we factor out 2 from the equation.

`y = 2(x^2+ 3/2x + 1/2)`

Now, take the coefficient of ‘x’, which is `3/2`, divide it by 2, whence we get `3/4` and square it, which gives us `(3/4)^2`. Add and subtract `(3/4)^2` from the quadratic expression inside the parenthesis.

`y = 2(x^2 + 3/2x + (3/4)^2 - (3/4)^2 + 1/2)`

Compare the above highlighted part with `a^2 + 2ab + b^2`. Thus we get `a = x` and `b = 3/4`. Now apply the formula `a^2 + 2ab + b^2 = (a + b)^2`.

`y = 2((x + 3/4)^2 - (3/4)^2 + 1/2)`

Simplify,

`y = 2((x + 3/4)^2 - 1/16)`

Remove the parenthesis by multiplying with 2,

`y = 2(x + 3/4)^2 – 1/8`

The above equation is now in vertex form.

Step 2: Identify ‘a’, ‘h’ and ‘k’

Comparing the equation `y = 2(x + 3/4)^2 – 1/8` with the vertex form `y = a(x – h)^2 + k` we get

`a = 2`
`h = -3/4`
`k = -1/8`

Step3: Focus of the parabola is `(h, k + 1/(4a))`

Plug the values of ‘a’, ‘h’ and ‘k’ into the formula for focus to get its coordinates:

`(-3/4, -1/8 + 1/(4*2))`

Simplify,

`(-3/4, 0)`

Thus, the coordinates of the focus of the parabola `y = 2x^2 + 3x + 1` are `(-3/4, 0)`.