Identify the vertex, focus, equation of axis of symmetry, equation of directrix, direction of opening, vertex, and length of the latus rectum (focal width) of 2y² + x + 20y + 51 = 0 .
sorry - surely you can take care of the cosmetics.
2y^2 + x + 20y + 51 = 0
-2(y^2+10y+25) = x+1
(y+5)^2 = -1/2 (x+1)
Now recall that the parabola
y^2 = 4px has
center at (0,0)
directrix at x = -p
focus at (p,0)
So, yours has been shifted so the vertex is at (-1,-5)
yours has p = -1/8, so
directrix x = -1 + 1/8 = -7/8
latus rectum = 4p = 1/2
directrix is the line x = -p = 1/8
it opens to the left, since the focus is at -1 - 1/8 = -9/8
I appreciate your answers, but at the same time, I'm sorry for not saying this, but put the answers like this:
A) Focus:
(solution here.)
---------------------------
B) Vertex:
(solution here.)
---------------------------
and so on
To identify the vertex, focus, equation of the axis of symmetry, equation of the directrix, direction of opening, vertex, and length of the latus rectum (focal width) of the given equation 2y² + x + 20y + 51 = 0 (in the form of Ax² + Bxy + Cy² + Dx + Ey + F = 0), we can follow these steps:
Step 1: Rearrange the equation
We need to rearrange the given equation in a standard form: y² = 4ax + b, where (h, k) is the vertex, a is the distance from the vertex to the focus, and b is the distance between the vertex and the directrix.
2y² + x + 20y + 51 = 0
Rearranging the equation, we get:
2y² + 20y + x = -51
Step 2: Complete the square
To complete the square, we group the terms involving y and the constant term separately:
(2y² + 20y) + x = -51
Next, we take the coefficient of y (which is 20), divide it by 2, and then square it: (20/2)² = 100.
We add and subtract 100 inside the parenthesis:
(2y² + 20y + 100 - 100) + x = -51
Step 3: Simplify and factor
Now, we simplify the quadratic expression inside the parentheses:
2(y² + 10y + 50) + x = -51
Factor the expression inside the parentheses:
2(y + 5)² + x = -51
Step 4: Rewrite the equation in standard form
To get the equation in the standard form of y² = 4ax + b, we divide the whole equation by 2:
(y + 5)² = (-1/2)x - 51/2
Step 5: Identify the values
Comparing the equation with the standard form y² = 4ax + b, we can identify the values:
Vertex: (-5, 0) (h = -5, k = 0)
Focus: (a, k) = (-1/8, 0)
Equation of the axis of symmetry: x = h = -5
Equation of the directrix: x = -a (x = 1/8)
Direction of opening: Horizontal
Length of latus rectum (focal width): 4a = -1/2
So, the vertex is (-5, 0), the focus is (-1/8, 0), the equation of the axis of symmetry is x = -5, the equation of the directrix is x = 1/8, the direction of opening is horizontal, and the length of the latus rectum (focal width) is 1/2.