How To Find Focus Of Parabola

A parabola is a curve that is shaped like the letter “U” or “C”. It is defined as the set of all points that are equidistant from a fixed line called the directrix and a fixed point called the focus.

Before we delve into how to find the focus of a parabola, it is essential to understand the basic properties of a parabola. The focus of a parabola is a fixed point located on the axis of symmetry, equidistant from each point on the parabola. It plays a crucial role in defining the shape and nature of the parabolic curve.

When it comes to the focus of a parabola, there are two types of parabolas that we commonly encounter:

• Vertical Parabola: When the parabola opens vertically, its general equation is of the form $y = ax^2 + bx + c$.
• Horizontal Parabola: When the parabola opens horizontally, its general equation is of the form $x = ay^2 + by + c$.

Now, let’s discuss how to find the focus of a parabola. The approach to determining the focus varies for vertical and horizontal parabolas. We will explore both cases in detail:

Vertical Parabola

For a vertical parabola, the focus lies at a point $F(h, k + \frac{1}{4a})$. To find the focus, follow these steps:

1. Identify the values of a, b, and c: from the equation of the parabola $y = ax^2 + bx + c$.
2. Determine the coordinates of the focus: using the formula $F(h, k + \frac{1}{4a})$, where (h, k) is the vertex of the parabola.
3. Substitute the known values: Calculate the focus coordinates by substituting the values of a, h, and k into the formula.

Horizontal Parabola

For a horizontal parabola, the focus lies at a point $F(h + \frac{1}{4a}, k)$. Here’s how you can find the focus:

1. Find the values of a, b, and c: from the equation of the parabola $x = ay^2 + by + c$.
2. Calculate the focus coordinates: using the formula $F(h + \frac{1}{4a}, k)$, where (h, k) represents the vertex of the parabola.
3. Substitute the known variables: plug in the values of a, h, and k into the formula to determine the focus coordinates.

To solidify your understanding, let’s consider a couple of examples for finding the focus of a parabola:

Example 1: Vertical Parabola

Consider the vertical parabola given by the equation $y = 2x^2 + 4x + 6$. Determine the focus of this parabola.

1. Identify the values of a, b, and c: $a = 2, b = 4, c = 6$.
2. Calculate the vertex coordinates: $h = -\frac{b}{2a} = -\frac{4}{4} = -1, k = c – ah^2 = 6 – 2(-1)^2 = 4$.
3. Determine the focus: $F(-1, 4 + \frac{1}{4(2)}) = F(-1, 4.5)$.
4. The focus of the parabola is: $(-1, 4.5)$.

Example 2: Horizontal Parabola

Let’s take the horizontal parabola given by the equation $x = 3y^2 – 6y + 9$. Find the focus of this parabolic curve.

1. Determine the values of a, b, and c: $a = 3, b = -6, c = 9$.
2. Compute the vertex coordinates: $h = -\frac{b}{2a} = -\frac{-6}{2(3)} = 1, k = c – ah^2 = 9 – 3(1)^2 = 6$.
3. Find the focus: $F(1 + \frac{1}{4(3)}, 6) = F(1.083, 6)$.
4. The focus of the parabola is: $(1.083, 6)$.

Understanding how to find the focus of a parabola is crucial for analyzing and graphing parabolic functions accurately. Whether dealing with vertical or horizontal parabolas, knowing the focus provides valuable insight into the geometry of the curve. By following the systematic steps outlined in this guide, you can easily determine the focus of any parabolic equation.