Answer:
To find the focus and directrix of the parabola given by the equation:
y^2 - 4x - 8y + 36 = 0
We first need to rewrite the equation in the standard form for a parabola with a vertical axis of symmetry:
(y - k)^2 = 4a(x - h)
Where (h, k) is the vertex of the parabola, and 'a' is the distance between the vertex and the focus (and also between the vertex and the directrix).
Let's complete the square to rewrite the equation in the standard form:
y^2 - 8y + 36 - 4x = 0
First, move the constant term to the other side:
y^2 - 8y = 4x - 36
Now, complete the square for the y terms. To do this, we need to add and subtract the square of half of the coefficient of y (which is -8/2 = -4):
y^2 - 8y + (-4)^2 = 4x - 36 + (-4)^2
y^2 - 8y + 16 = 4x - 20
Now, rewrite the equation:
(y^2 - 8y + 16) = 4(x - 5)
Now the equation is in the standard form. We can see that the vertex is at the point (h, k) = (5, 4), and 'a' is 1 (from the coefficient of 4 in front of the right side).
So, the vertex is (5, 4), and 'a' is 1. Now we can find the focus and directrix.
1. Focus (F):
The focus is located a distance 'a' units to the right of the vertex along the axis of symmetry. In this case, it is 1 unit to the right of the vertex (since 'a' is 1).
So, the focus (F) is at (5 + 1, 4) = (6, 4).
2. Directrix (D):
The directrix is located a distance 'a' units to the left of the vertex along the axis of symmetry. In this case, it is 1 unit to the left of the vertex (since 'a' is 1).
So, the equation of the directrix (D) is the vertical line x = 5 - 1 = 4.
Therefore, the focus of the parabola is at (6, 4), and the directrix is the vertical line x = 4.
Explanation: