Determine the focus of the parabola with the equation x2 - 6x + 5y = -34
x2 - 6x + 5y = -34
5y = -x^2 + 6x - 34
= -1(x^2 - 6x + 9 - 9) -34
= -(x-3)^2 + 9 - 34
y = (-1/5)(x - 3)^2 - 25/5
y = (-1/5)(x-3)^2 - 5
lots of algebraic typing ahead, so I will let you watch 4 great videos dealing with this topic by the remarkable Mr. Khan
https://www.khanacademy.org/math/precalculus/conics-precalc/focus-and-directrix-of-a-parabola/v/focus-and-directrix-introduction
the first 3 are a good introduction to your topic, and the last one deals with our question.
you should get:
http://www.wolframalpha.com/input/?i=focus+of+x%5E2+-6x%2B5y+%3D+-34
moving right along, you know that the parabola
x^2 = 4py
has focus at (0,p). You have
(x-3)^2 = -5(y+5)
So, 4p = -5 and the focus is at (0,-5/4) but shifted by (3,-5) to
(3,-25/4)
as Reiny's graph shows.
To determine the focus of a parabola, we can rewrite the equation in standard form, which is in the form (x-h)^2 = 4p(y-k), where (h, k) represents the vertex of the parabola.
Let's begin by rearranging the given equation:
x^2 - 6x + 5y = -34
Adding 34 to both sides:
x^2 - 6x + 5y + 34 = 0
Now, we need to complete the square to convert it into the standard form.
Take half of the coefficient of x (which is -6), square it (-6/2)^2 = 9, and add it to both sides:
x^2 - 6x + 9 + 5y + 34 = 9
(x^2 - 6x + 9) + 5y + 34 = 9
Rearranging the terms:
(x - 3)^2 + 5y + 34 = 9
Subtracting 34 from both sides:
(x - 3)^2 + 5y = -25
Now, we can identify the vertex and the value of p.
The vertex of the parabola is (h, k), which is (3, -25/5).
The value of p is the distance from the vertex to the focus.
Since the parabola opens upward, we know that p is positive.
Let's rewrite the equation using the vertex form:
(x - h)^2 = 4p(y - k)
(x - 3)^2 = 4p(y + 5)
Comparing this with the given equation, we can see that 4p = 5. Thus, p = 5/4.
Therefore, the focus of the parabola is located at (3, -25/5 + 5/4) or (3, -25/5 + 25/20), which simplifies to (3, -5/4).
To determine the focus of the parabola, we need to rewrite the given equation in vertex form, which is of the form (x - h)^2 = 4p(y - k), where (h, k) represents the vertex and p represents the distance from the vertex to the focus.
Let's begin by rearranging the given equation:
x^2 - 6x + 5y = -34
To complete the square, we need to isolate the x and y terms:
x^2 - 6x = -5y - 34
Now let's focus on completing the square for the x terms. We take half of the coefficient of x (-6/2 = -3) and add its square (3^2 = 9) to both sides of the equation:
x^2 - 6x + 9 = -5y - 34 + 9
(x - 3)^2 = -5y - 25
Now we can rewrite the equation in vertex form by dividing through by -5:
-1/5 * (x - 3)^2 = y + 5
Comparing the vertex form equation to the standard equation form (x - h)^2 = 4p(y - k), we can see that h = 3 and k = -5.
Therefore, the vertex of the parabola is (3, -5).
To find the focus, we need to determine the value of p. In the vertex form, p is the distance from the vertex to the focus, and it is calculated as p = 1 / (4a), where a is the coefficient of the y term.
In our equation, the coefficient of y is -1/5. Therefore, a = -1/5, and we can calculate p as follows:
p = 1 / (4 * (-1/5))
p = 1 / (-4/5)
p = -5/4
So, the focus of the parabola is located at (h, k + p), which in this case is (3, -5 - 5/4).
To simplify, let's get a common denominator for the fraction:
-5 - 5/4 = -20/4 - 5/4 = -25/4
Hence, the focus of the parabola with the equation x^2 - 6x + 5y = -34 is located at (3, -25/4).