the line with the equation x+y=3 is graphed on the same xy-plane as the parabola with vertex (0,0) and focus (0,3). What is the point of intersection of the two graphs?

To find the point of intersection between the line and the parabola, you need to solve the system of equations formed by the line equation and the equation of the parabola.

1. Line equation: x + y = 3
2. Equation of the parabola: (x - h)^2 = 4p(y - k)

Given:
- Vertex of the parabola: (h, k) = (0, 0)
- Focus of the parabola: (h, k + p) = (0, 3)

From the properties of the parabola, we know that the distance between the focus and the vertex is equal to the distance between the vertex and the directrix. In this case, the focus is (0, 3), and the vertex is (0, 0). Therefore, the distance between them is 3 units.

We also know that the equation of the directrix is given by y = k - p. Substituting the vertex coordinates into the equation, we find that the directrix is y = 0 - (-3) = 3.

So, the parabola has a focus at (0, 3) and a directrix at y = 3. Since the line and the parabola intersect, the point of intersection must lie on both the line and the parabola.

By substituting y = 3 - x into the equation of the parabola, we can find its x-coordinate:

(x - h)^2 = 4p(y - k)
(x - 0)^2 = 4p(3 - x - 0)
x^2 = 12p - 4px

Substituting the value of y from the line equation into the above equation, we can find the x-coordinate:

x^2 = 12p - 4px
x^2 = 12p - 4x(3 - x)
x^2 = 12p - 12x + 4x^2
3x^2 - 12x + 12p = 0

Next, we solve this quadratic equation to find the x-coordinate(s) of the point(s) of intersection. If there are two solutions, we substitute them back into the line equation to find the corresponding y-coordinates.

Finally, we have the point(s) of intersection as (x, y) coordinates, which are the solutions to the system of equations formed by the line and the parabola.

The School Subject is not 11th grade. If it is Math, please state that.

Sra