Find the focus and directrix of the parabola: x=(1/8y^2)-(1/2y)+(3/2)
To find the focus and directrix of the given parabola, we can start by rewriting the equation in a standard form.
The given equation is x = (1/8)y^2 - (1/2)y + (3/2).
To put this equation in standard form, we move the x-term to the right-hand side:
x - (1/8)y^2 + (1/2)y = (3/2).
Now, in order to complete the square, we need to factor out the coefficient of y^2 from the first two terms, and also factor out the coefficient of y from the last two terms:
-(1/8)(y^2 - 4y) = x - (1/2)y + (3/2).
Next, we need to find the value that completes the square for the y-term. To do this, we take half of the coefficient of y, square it, and add it to both sides:
-(1/8)(y^2 - 4y + 4) = x - (1/2)y + (3/2) + (1/8)(4).
Simplifying this equation, we have:
-(1/8)(y - 2)^2 = x - (1/2)y + (1/2).
To eliminate the coefficient on the left-hand side, we can multiply both sides by -8:
(y - 2)^2 = -8x + 4y - 4.
Expanding the square on the left-hand side, we get:
(y^2 - 4y + 4) = -8x + 4y - 4.
Rearranging this equation, we have:
y^2 - 4y + 8x - 4y = 0.
Combining like terms, we get:
y^2 - 8y + 8x = 4.
Now, the equation is in the standard form: y^2 = 4px.
Comparing this with the standard form of a parabola, we can see that the coefficient of x is 8, which means that the value of p is 1/8.
The focus of a parabola is given by (p, 0), where p is the distance from the focus to the vertex along the x-axis. In this case, the focus is located at (1/8, 0).
The directrix of a parabola is a line perpendicular to the x-axis and p units away from the vertex. The equation of the directrix is x = -p. Plugging in the value of p, we get x = -1/8.
Therefore, the focus of the given parabola is (1/8, 0) and the directrix is x = -1/8.