This is a common question that we get from two audiences. It is most commonly asked by algebra students who want to calculate the point in the plane, when the equation of the parabola is given. More advanced students sometimes must prove this formula from the definition of the focus. We will handle this scenario second. We have also included the process for finding the focus of a parabola when we’re only given the vertex and a point on the parabola.
Formula for the Focus
We will assume you are given the vertex form of the parabola. Say you have y = 3(x-1)² + 2. We are asked to find the focus. How do we go about it?
The formula we need in this case is F(h, k+p). Recall that the vertex is V(h,k) when y = a(x-h)²+k. Because we have the equation in vertex form already, we know the vertex is at V(1,2). The only think we need now is p, where p = 1/(4a). As a = 3, in this case, p = 1/(4*3) = 1/12. So F(h,k+p) = (1, 2+1/12) = (1, 25/12). And so, the focus is the point F(1, 25/12).
Thus, to find the focus of the parabola, we (1) find the vertex, V(h,k); then (2) compute p = 1/(4a); then (3) compute F(h,k+p).
Here’s another example.
Example 1: Find the focus of the parabola y = -2(x+4)²-1.
Solution: Because y = -2(x+4)²-1, the vertex is the point V(-4,-1). Next we calculate p = 1/(4a), noting that a = -2, in this case. When we do we find p = -1/8. Finally the focus is the point (-4, -1 + -1/8) = (-4, -9/8). ■
Sometimes we are given the equation of the parabola in standard form and must either convert the equation to vertex form or use formulas to compute the vertex.
Example 2: Find the focus of the parabola y = (1/2)x² – 5.
Solution: Note that, in this example b = 0, and that any time that b = 0, the standard form and vertex form of the equation are identical. But in this case, we will compute the vertex using the formulas we would use if the vertex form of the equation were not also given to us.
The vertex formula gives us h = -b/(2a) and k = (4ac-b²)/(4a). In this case, a = 1/2, b = 0, and c = -5. Thus h = 0 and k = -5 and the vertex is the point (0, -5). With a = 1/2, then p = 1/2, so that F is the point (0, -5+1/2) = (0, -9/2). ■
Deriving the Formula for the Focus
For more advanced students who wish to prove the formula for the focus, here is an outline of one way to do it.
HOW TO DERIVE THE FORMULA FOR THE FOCUS:
Start with the definition of a parabola. A parabola is the set of all points equidistant from a point F, called the focus to a point on a line, called the directrix. Let F be the point (m,n), and the directrix be the line y = t. Now pick an arbitrary point (x,y). We use the fact that his point must be equidistant from F and line D, such that the distance formula from F to P must be the same as P to D. When we equate the distance, we look for a relationship between x and y. What we find is that y is quadratic in x.
Next, because we know y is quadratic in x, write the coefficients of this quadratic as A, B, and C, and set out to write m, n, and t in terms of A, B and C. We encounter a system of three equations in three unknowns. Once we solve each m, n and t, in terms of A, B, and C, we obtain the usual relationships by defining a point V(h,k), called the vertex, where h = -B/(2A), k = (4AC-B²)/(4A). Then if p = 1/(4A), then focus, F(m,n) is given by m = h; n = k+p, and directrix is given by y = k-p.
Finding the Focus, Given the Vertex and Point on the Parabola
When we’re given the vertex and a point on the parabola, and asked to find the focus, we’re going to want to use a the following process. First, we write the equation of the parabola in vertex form, to the extent we can. We will be able to write the entire vertex form of the equation except for our leading coefficient, a. Next, we’ll use the point that we’re given as a solution to this equation. This means we can substitute the point (x,y) in for the x and y values in the vertex equation. Then we solve for a. After we have a, we can compute p = 1/(4a), and F(h,k+p), which will give us our focus.
Here’s an example of how we go about it.
Example: Given the vertex is at V(1,2) and the point (3,4) is on the parabola, find the coordinates of the focus.
Solution: First use the vertex to write the equation of the parabola in vertex form to the fullest extent we can. Recall vertex form of the parabola is y = a(x-h)²+k. Because we are given the vertex is V(1,2), the vertex form of the equation of our parabola is y = a(x-1)² + 2.
Next, we use the fact that the point (3,4) is a solution to this equation to find the value of a. We do this by plugging the values x=3 and y=4 into our equation: 4 = a(3-1)²+2. Then we solve to find that a=1/2.
Because we now have the value of a, we can easily find p=1/(4a)=1/2. So, F(h,k+p) = F(1,2+1/2)=F(1,5/2), and our focus is the point F(1,5/2). ■
Did we answer your question about finding the focus of a parabola? Share your thoughts in the comments.