Find the coordinates of the focus, the lenth of the lactus rectum and coordinates of its endpoints and the equation of the directrix, then sketch each curve of 2y^2=8x.
You have
y^2 = 4x
This is just the standard equation
y^2 = 4px
with p=1. Check your book or online for properties of parabolas. You can start here if you wish:
http://www.wolframalpha.com/input/?i=parabola+y%5E2+%3D+4x
To find the coordinates of the focus, the length of the latus rectum, the coordinates of its endpoints, and the equation of the directrix for the curve 2y^2 = 8x, we can begin by analyzing the equation.
The given equation, 2y^2 = 8x, is a quadratic equation in standard form, where the coefficient of x is equal to 8. It is a parabolic equation that opens to the right. The general form of a parabolic equation is x = (1/4a)y^2 for the parabola that opens to the right.
Comparing this general form to the given equation, we can determine that a = 1/32. The focus of the parabola is located at a distance of a units to the right of the vertex. Since the vertex is at the origin (0, 0), the focus will be at (a, 0).
The focus coordinates can be determined by substituting the value of a into the equation. In this case, a = 1/32. Therefore, the focus coordinates are (1/32, 0).
The latus rectum is a line segment perpendicular to the axis of symmetry, passing through the focus, and whose endpoints lie on the parabola. The length of the latus rectum can be found using the formula 4a, where a is the constant we obtained earlier.
Substituting the value of a into the formula, we find that the length of the latus rectum is equal to 4 * (1/32) = 1/8.
The endpoints of the latus rectum can be calculated by adding or subtracting the length of the latus rectum from the x-coordinate of the focus. In this case, the endpoints are (1/32 + 1/8, 0) and (1/32 - 1/8, 0) which simplify to (3/32, 0) and (-7/32, 0), respectively.
To find the equation of the directrix, we can use the formula x = -a. Substituting the value of a (-1/32), we find that the equation of the directrix is x = -1/32.
Now, as for sketching the curve, we know that the vertex is at the origin (0, 0), the focus is at (1/32, 0), the length of the latus rectum is 1/8, and the directrix is x = -1/32. We can plot these key points on a graph and then sketch the parabolic curve that opens to the right.
I hope this explanation helps you find the coordinates of the focus, the length of the latus rectum, the coordinates of its endpoints, and the equation of the directrix for the given equation.