What is an equation of a parabola with the given vertex and focus?
vertex(-2,5)
focus(-2,6)
I forgot my answer:
(x+2)^2 = 4(y-5)
If you make a sketch, you will see that the directrix must be y = 4
let P(x,y) be any point
√( (y-6)^2 + (x+2)^2 ) = √((y-4)^2 + 0)
square and expand
y^2 - 12y + 36 + (x+2)^2 = y^2 - 8y + 16
(x+2)^2 = 4y - 20
(x+2)^2 = 4(y- 5)
I agree
Thank you
To find the equation of a parabola given the vertex and focus, we can use the formula:
1. Determine the direction of the parabola based on the vertex and focus.
- If the focus is above the vertex, the parabola opens upwards. If the focus is below the vertex, the parabola opens downwards.
2. Determine the distance between the vertex and focus. This distance is called the "focal length" (f).
3. Use the vertex and the focal length to write the equation of the parabola in standard form.
In this case, the vertex is given as (-2, 5) and the focus is given as (-2, 6).
Step 1: Determine the direction of the parabola
Since the focus is above the vertex, the parabola opens upwards.
Step 2: Determine the focal length (f)
The focal length is the vertical distance between the vertex and the focus. In this case, it is 1 because the y-coordinate of the focus is 6 and the y-coordinate of the vertex is 5.
Step 3: Write the equation of the parabola in standard form
The equation of a parabola in standard form is (x - h)^2 = 4p(y - k), where (h, k) is the vertex and p is the focal length.
Substituting the given values into the standard form equation, we get:
(x - (-2))^2 = 4(1)(y - 5)
(x + 2)^2 = 4(y - 5)
Therefore, the equation of the parabola with the given vertex and focus is (x + 2)^2 = 4(y - 5).