Answer:
Step-by-step explanation:
The directrix y=-6 is parallel to the x axes, so parabola has equation of the form
where is the vertex of parabola and p is parabola's parameter.
By the definition, is the distance from the vertex to the directrix, so
Hence, the equation of parabola is
See attached diagram for the graph of parabola and its directrix.
In vertex form this equation is
Answer:
The answer is circle; (x')² + (y')² - 4 = 0
Step-by-step explanation:
* At first lets talk about the general form of the conic equation
- Ax² + Bxy + Cy² + Dx + Ey + F = 0
∵ B² - 4AC < 0 , if a conic exists, it will be either a circle or an ellipse.
∵ B² - 4AC = 0 , if a conic exists, it will be a parabola.
∵ B² - 4AC > 0 , if a conic exists, it will be a hyperbola.
* Now we will study our equation:
* 2x² + 2y² = 8
∵ A = 2 , B = 0 , C = 2
∴ B² - 4AC = (0) - 4(2)(2) = -16 < 0
∵ B² - 4AC < 0
∴ it will be either a circle or an ellipse
* Lets use this note to chose the correct figure
- If A and C are equal and nonzero and have the same sign,
then the graph is a circle.
- If A and C are nonzero, have the same sign, and are not equal
to each other, then the graph is an ellipse.
∵ A = 2 and C = 2
∴ The graph is a circle.
∵ D and E = 0
∴ The center of the circle is the origin (0 , 0)
∵ Ф = 30°
∴ The point (x , y) will be (x' , y')
- Where x = x'cosФ - y' sinФ and y = x'sinФ + y'cosФ
∴ x = x'cos(30°) - y'sin(30°)
∴ y = x'sin(30°) + y'cos(30°)
∴ x = (√3/2)x' - (1/2)y' and y = (1/2)x' + (√3/2)y'
∴
∴
* Lets substitute x and y in the first equation
∴
* Use the foil method
∴
* Open the brackets
∴
* Collect the like terms
∴
* Simplify the fraction
∴ 2(x')² + 2(y')²= 8
* Divide each side by 2
∴ (x')² + (y')² = 4
∴ The equation of the circle is (x')² + (y')² = 4
* The general equation of the circle is (x')² + (y')² - 4 = 0
after rotation 30° about the origin
* Look to the graph
- The blue circle for the equation 2x² + 2y² = 8
- The blue circle for equation (x')² + (y')² - 4 = 0
* That is because the two circles have same centers and radii
- The green line is x' and the purple line is y'
