1.
What is the focus, and directrix of the parabola: x^{2} = 28y?
Correct Answer
B. Focus: (0,7) Directrix: y=-7
Explanation
The parabola x^2 = 28y has a vertical axis of symmetry, which means that the focus and directrix will have the same x-coordinate. The equation of the parabola is in the form (x-h)^2 = 4p(y-k), where (h,k) is the vertex and p is the distance from the vertex to the focus and directrix. In this case, the vertex is (0,0) and p = 7. Since the vertex is at the origin, the focus will be at (0, p) = (0,7). The directrix will be a horizontal line p units below the vertex, so the equation of the directrix is y = -7.
2.
What is the standard form of the equation of the parabola whose focus is (0, 4) and a directrix is y=-4.
Correct Answer
A. Y = (1/16)x^{2}
Explanation
The standard form of the equation of a parabola is y = ax^2, where a is a constant. In this case, the focus is (0, 4) and the directrix is y = -4. Since the focus is above the directrix, the parabola opens upwards. The distance from the vertex to the focus is equal to the distance from the vertex to the directrix. The formula for the distance from the vertex to the focus is 1/(4a), and the formula for the distance from the vertex to the directrix is -1/(4a). By substituting the given values, we can solve for a and obtain a = 1/16. Therefore, the equation of the parabola is y = (1/16)x^2.
3.
The focus is always inside the parabola.
Correct Answer
A. True
Explanation
The statement is true because the focus of a parabola is a fixed point inside the curve. In a parabola, all points are equidistant from the focus and the directrix. This means that any line segment drawn from any point on the parabola to the focus will have the same length as the line segment drawn perpendicular to the directrix. Therefore, the focus is always located inside the parabola.
4.
(y-4)^{2} = -8(x+1)
What are the coordinates for the focus here?
Correct Answer
D. (-3,4)
Explanation
The equation given is that of a parabola in standard form, where the vertex is at (-1, 4). The focus of a parabola is located at a distance of p units from the vertex, where p is the distance between the vertex and the directrix. In this case, the directrix is a vertical line given by x = -3. Since the parabola opens to the right, the focus will be to the right of the vertex. Therefore, the coordinates for the focus are (-3, 4).
5.
What is the equation of the directrix here?
(x+3)^{2} = 4(y+5)
Correct Answer
D. Y=-6
Explanation
The equation of the directrix is y = -6. This can be determined by comparing the given equation to the standard form of a parabola, which is (x-h)^2 = 4p(y-k), where (h,k) is the vertex and p is the distance between the vertex and the focus/directrix. In the given equation, the vertex is (-3,-5) and the distance between the vertex and the directrix is 6 units below the vertex. Therefore, the equation of the directrix is y = -6.
6.
What is the equation of the directrix?
(x+3)^{2} = 4(y+5)
Correct Answer
C. Y=-6
Explanation
The equation of the directrix for a parabola in vertex form is given by y = k - p, where (h, k) is the vertex and p is the distance between the vertex and the focus. In this case, the vertex is (-3, -5) and the distance between the vertex and the focus is 1. Therefore, the equation of the directrix is y = -5 - 1 = -6.
7.
Which way does the parabola open?
12(x - 6)^{ }= y^{ 2}
Correct Answer
B. Right
Explanation
The equation of the parabola is in the form of (x-h)^2 = 4p(y-k), where (h,k) is the vertex of the parabola and p is the distance from the vertex to the focus. In this case, the equation is in the form of (x-6)^2 = y/12. Since the coefficient of y is positive, the parabola opens upwards. Therefore, the correct answer is "Up".
8.
What is the directrix of the parabolic conic section y=x^{2}?
Correct Answer
A. Y=-1/4
Explanation
The directrix of a parabolic conic section is a line that is equidistant from the focus and all points on the parabola. In this case, the equation of the parabola is y=x^2, which opens upwards. Since the coefficient of x^2 is positive, the parabola opens towards the positive y-axis. Therefore, the directrix will be a horizontal line that is below the vertex of the parabola. The equation y=-1/4 represents a horizontal line that is 1/4 units below the vertex, which satisfies the conditions for the directrix.
9.
Use the distance formula to write the equation in standard form given the focus is (0,3) and the directrix is at y=5.
Correct Answer
C. -4(y-4) = x^{2}
Explanation
The given equation -4(y-4) = x2 represents a parabola in standard form. The standard form of a parabola equation is (y-k) = a(x-h)2, where (h,k) is the vertex of the parabola. In this case, the vertex is (0,3) as given. Therefore, the correct equation in standard form is (y-3) = (-1/4)x2 or -4(y-3) = x2.
10.
A parabola is the set of all points equidistant from the focus and the directrix.
Correct Answer
A. True
Explanation
A parabola is a curve that is symmetric and U-shaped. The given statement is true because it accurately describes the definition of a parabola. The focus is a fixed point inside the parabola, and the directrix is a fixed line outside the parabola. The parabola is formed by all the points that are equidistant from the focus and the directrix. This property is what distinguishes a parabola from other types of curves. Therefore, the statement is correct.