Find the focus, directrix, and focal diameter of the parabola.
y2 = 8x
This is an X-parabola.
Y^2 = 8x.
X = (1/8)y^2
k = Yv = -b/2a = 0/(1/4) = 0
h = Xv = (1/8*0^2 = 0.
V(h,k) = V(0,0).
D(X1,K), V(h,k), F(X2,K).
D(X1,0), V(0,0), F(X2,0).
DV = 0-X1 = 1/4a.
0-X1 = 1/(1/2) = 2.
X1 = -2.
VF = X2-0 = 1/4a
X2 = 1/(1/2) = 2
Focal Dia.(Ver. line).
A(2,Y1), F(2,0), B(2,Y2).
a = 1/8
1/a = 8 = Focal Dia.
AF = 0-Y1 = 8/2 = 4.
Y1 = -4.
FB = Y2-0 = 8/2 = 4
Y2 = 4.
Why did the math book go to the therapist? Because it had too many problems! But don't worry, I'll help you solve this one.
The given equation is y^2 = 8x, which is in the standard form for a parabola. By comparing it to the general equation (y - k)^2 = 4a(x - h), we can see that h = 0, k = 0, and a = 2.
First, let's find the focus of the parabola. Since the value of a is positive, the parabola opens to the right. The focus is located at (h + a, k), which in this case is (0 + 2, 0) = (2, 0).
Next, let's find the directrix. The directrix is a vertical line located at x = (h - a), which in this case is (0 - 2) = (-2). Therefore, the equation of the directrix is x = -2.
Lastly, let's find the focal diameter. The focal diameter is the distance between the focus and a point on the parabola that lies on a line parallel to the directrix. Since the parabola opens to the right, we can use the distance formula to find the focal diameter.
Let's take a point P on the parabola with coordinates (x, y). The distance between P and the focus (2, 0) would be equal to the perpendicular distance between P and the directrix x = -2.
Using the distance formula, we can set up the equation:
√((x - 2)^2 + (y - 0)^2) = |x - (-2)|
Simplifying, we have:
√(x^2 - 4x + 4 + y^2) = |x + 2|
Squaring both sides, we get:
x^2 - 4x + 4 + y^2 = (x + 2)^2
Simplifying further:
x^2 - 4x + 4 + y^2 = x^2 + 4x + 4
Canceling out common terms, we get:
-4x + y^2 = 4x
Rearranging, we have:
y^2 = 8x
Wait a minute... This is the same equation as the given one! That means that every point on the parabola satisfies this equation! So, the focal diameter of the parabola is infinite.
So, to recap:
- The focus of the parabola is (2, 0).
- The directrix of the parabola is x = -2.
- The focal diameter of the parabola is infinite.
Hope that solved your problem, and remember, math problems can be a bit parabolic, but they always have a solution in the end!
To find the focus, directrix, and focal diameter of the parabola given by the equation y^2 = 8x, we can use the standard form of a parabola equation:
4p(y - k) = (x - h)^2
where (h, k) is the vertex and p is the distance from the vertex to the focus or directrix.
Comparing this standard form to the given equation, we can see that h = 0 and k = 0.
1. Focus:
The focus is given by the formula (h + p, k), so in this case, it becomes (0 + p, 0). Since p is not directly given, we can rearrange the given equation to determine the value of p.
y^2 = 8x
We can rewrite the equation as:
x = y^2/8
Comparing this to the standard form equation, we have h = 0 and k = 0. Therefore, the vertex is (0, 0).
Now, we know that the distance from the vertex to the focus is p, so we can substitute (0, 0) into the equation:
0 = (0^2)/(8p)
This simplifies to:
0 = 0
Since the equation simplifies to 0 = 0, we can't determine the exact value of p. However, we know that the focus is at (h + p, k), so the focus is at (0, 0).
2. Directrix:
The directrix is given by the equation y = k - p. In this case, k = 0. Since p can't be determined exactly, we cannot find the equation of the directrix.
3. Focal Diameter:
The focal diameter is the distance between two points on the parabola that are equidistant from the focus. In this case, the vertex is the point on the parabola that is equidistant from the focus. Therefore, the focal diameter is the distance between the vertex (0, 0) and the focus (0, 0), which is 0.
In summary:
- Focus: (0, 0)
- Directrix: Cannot be determined
- Focal Diameter: 0
To find the focus, directrix, and focal diameter of the given parabola equation, we need to convert it into the standard form of the parabola equation. The standard form is (x - h)^2 = 4p(y - k), where (h, k) is the vertex of the parabola and p is the distance between the vertex and the focus.
In the given equation, y^2 = 8x, we have a positive coefficient for x, which means the parabola opens to the right. By comparing it with the standard form, we can see that h = 0 and k = 0.
To find p, we need to rewrite the equation in the standard form: (x - 0)^2 = 4p(y - 0), which simplifies to x^2 = 4py.
Now, we can see that p = 2 because the coefficient of y is 1/4p (i.e., 1/4p = 1/4(2) = 1/8).
The vertex, in this case, is at (h, k) = (0, 0). So the focus is located at (h + p, k) = (0 + 2, 0) = (2, 0).
The directrix is a vertical line located at y = k - p, which means it will be located at y = 0 - 2, which simplifies to y = -2.
The focal diameter is the distance between the focus points on the parabola, which is twice the distance between the vertex and the focus. So the focal diameter in this case is 2p = 2(2) = 4.
To summarize:
Focus: (2, 0)
Directrix: y = -2
Focal Diameter: 4