i need help with this equation anything help will be greatly appreciated
find the standard equation of a parabola with a focus (-2,0) directrix x=3
Determine the directrix of the parabola with the equation x2 - 6x + 5y = -34. Answer in complete equation.
(For example: y = 5)
To find the standard equation of a parabola with a given focus and directrix, we can use the formula:
1. Identify the vertex: The vertex is the midpoint between the focus and the directrix. In this case, the focus is (-2, 0) and the directrix is x = 3. So, the vertex can be found by taking the average of the x-coordinates and y-coordinates:
Vertex = ((-2 + 3) / 2, 0) = (0.5, 0)
2. Determine the distance from the vertex to the focus (also known as the focal length). Since the directrix is a vertical line, we can determine it as the absolute difference between the x-coordinate of the focus and the x-coordinate of the vertex:
Focal length = abs(-2 - 0.5) = abs(-2.5) = 2.5
3. Write the standard equation of the parabola: The standard equation of a parabola with a vertical axis is given by:
(x - h)^2 = 4p(y - k)
where (h, k) is the vertex and p is the focal length.
Plugging in the values we found:
(x - 0.5)^2 = 4 × 2.5(y - 0)
Simplifying, we get:
(x - 0.5)^2 = 10(y - 0)
Expand further if required.