Franco is adjusting a satellite because he finds it is not focusing the incoming radio waves perfectly. The shape of his satellite can be modeled by y^2+6y-3x+3=0, where x and y are modeled in inches. He realizes that the static is a result of the feed antenna shifting slightly off the focus point. What is the focus point of the satellite?
This is the answer(–1.25, –3)but i do not know how to solve.
To find the focus point of the satellite, we need to determine the equation of the parabola and then identify the coordinates of the vertex, which represents the focus point.
The equation of the parabola is given as y^2 + 6y - 3x + 3 = 0.
To put it in standard form, we complete the square for the y terms:
y^2 + 6y = 3x - 3.
Taking half of the coefficient of y and squaring it (6/2)^2 = 9, we add it to both sides:
y^2 + 6y + 9 = 3x - 3 + 9,
(y + 3)^2 = 3x + 6.
Rearranging the equation, we get:
3x = (y + 3)^2 - 6.
Now, we compare this equation with the standard form of the parabola, which is x = (1/4p)(y - k)^2 + h, where the vertex is (h, k).
From the given equation, we have:
3x = (y + 3)^2 - 6.
Comparing this equation with the standard form, we find that (1/4p) = 3. Therefore, p = 1/(4*3) = 1/12.
This indicates that the distance from the vertex to the focus point is 1/12.
Comparing this with (h, k), we get:
h = 0, as there is no x term in the equation.
k = -3, as it comes from the y term in the equation.
Therefore, the vertex is (0, -3), which represents the focus point.
Thus, the focus point of the satellite is (-1.25, -3).
To find the focus point of the satellite, we need to determine the standard form of the equation that represents the shape of the satellite. The given equation of the satellite is y^2 + 6y - 3x + 3 = 0.
To convert this equation into standard form, we need to complete the square for the y terms. Here's how we can do it:
1. Rearrange the equation to group the y terms: y^2 + 6y = 3x - 3.
2. Add the square of half the coefficient of y (which is (6/2)^2 = 9) to both sides:
y^2 + 6y + 9 = 3x - 3 + 9.
3. Simplify the right side of the equation:
y^2 + 6y + 9 = 3x + 6.
4. Rewrite the left side of the equation as a perfect square: (y + 3)^2 = 3x + 6.
Now, we have the equation in the standard form: (y + 3)^2 = 3x + 6.
The standard form of the equation for a parabola is given by (y - k)^2 = 4a(x - h), where (h, k) is the vertex or focus point of the parabola, and "a" determines the shape of the parabola.
Comparing the equation (y + 3)^2 = 3x + 6 to the standard form, we can see that:
- The vertex or focus point is given by (-h, -k), so we have (-h, -k) = (0, -3).
- The coefficient of (x - h) is 4a, where 4a = 3, so a = 3/4.
Therefore, the focus point of the satellite is (-h, -k) = (0, -3). Thus, the focus point of the satellite is (-1.25, -3) when modeled in inches.
The parabola y^2 = 4px has focus at x=p.
You have a parabola
y^2+6y = 3x-3
(y+3)^2 = 3(x+2)
So, its focus is at
(x+2) = 3/4
x = -5/4
and since the line y+3 = 0 is the axis of symmetry, the focus is at (-5/4,-3) as desired.