## Pages

### 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).