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?
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Sra
To find the point of intersection between the line and the parabola, you need to solve the system of equations formed by their equations.
1. Let's start with the equation of the line: x + y = 3.
2. Next, we need to determine the equation of the parabola with the given vertex and focus information. The vertex of the parabola is (0,0), which means its equation can be written in the form y = a(x - h)^2 + k, where (h, k) is the vertex. Since the vertex is (0,0), the equation simplifies to y = ax^2.
3. To find the value of 'a', we need to use the focus information. The focus of the parabola is given as (0,3), which means the distance from any point on the parabola to the focus should be equal to the perpendicular distance from that point to the directrix. In this case, the directrix is the line y = -3.
4. Using the formula for the distance between a point (x, y) and a line ax + by + c = 0, we can calculate the distance from any point on the parabola (x, ax^2) to the directrix y = -3.
The distance formula is: distance = |ax + by + c| / sqrt(a^2 + b^2).
Plugging in the values, we have:
distance = |x - (-3)| / sqrt(0^2 + 1^2)
= |x + 3|.
Since the distance from the focus to the parabola is equal to the distance from the parabola to the directrix, we can set up the equation:
|x + 3| = |y - 3|.
5. Now we have two equations:
x + y = 3,
|x + 3| = |y - 3|.
We can solve this system of equations by considering different cases. In this case, we'll consider:
x + 3 = y - 3, giving us x - y = -6,
and
x + 3 = -(y - 3), giving us x + y = 0.
6. Now we can solve the system of equations x + y = 3 and x + y = 0 using any method of solving simultaneous equations. Subtracting the second equation from the first, we get:
(x + y) - (x + y) = 3 - 0,
0 = 3.
Since the equation simplifies to 0 = 3, there are no solution points that satisfy both equations. Therefore, there is no point of intersection between the line and the parabola in the given xy-plane.