Respuesta :
1) We have that the equation is [tex]x^2=20y[/tex] , hence y=x^2/20. The standard equation of such an equation is y=[tex] \frac{1}{4p} x^2[/tex]. Hence, p=5 in this case. The focus is at (0,5) and the directrix is at y=-5 (a tip is that the directrix is always "opposite" the focus point of a parabola; if the directrix is at x=-7 for example, the focus is at (7,0)).
2) Similarly, we have that the equation is [tex]x=3y^2 \\ \frac{1}{4p} =3[/tex]. Thus, p=1/12. In this case, the parabola opens along the x-axis and the focus is at (1/12, 0). Also, the directrix is at x=-1/12. Hence the correct answer is B.
3) We are given that the parabola has a p of 9. Also, the focus lies along the y-axis, hence the parabola is opening along the y-axis. Finally, the focus is on the positive half, so the parabola is opening upwards. The equation for this case is y=[tex] y=\frac{1}{4p} x^2= \frac{1}{36 } x^2[/tex].
4) Similarly as above. The directrix is superfluous, we only need the p-value. THe same comments about the parabola apply and if we substitute p=8 in the formula: [tex]y= \frac{1}{4p} x^2 [/tex] we get y=[tex] \frac{1}{32} x^2[/tex].
5) This is somewhat different, even though we do not need the directrix again. The focus lies on the x-axis, thus the parabola opens in this direction. The focus lies on the positive part of the axis, thus the parabola opens to the right. We also are given p=7. Hence, the equation we need is of the form[tex]x= \frac{1}{4p} y^2[/tex]. Substituting p=7, we get [tex]x= \frac{1}{28} y^2[/tex].
6) The equation of a prabola with a vertex at (0,0) is of the form y=-ax^2. The minus sign is needed since the parabola is downwards. Since we are given anothe point, we can calculate a. We have to take y=-74 and x=14 feet (since left to right is 28, we need to take half). [tex]-a= \frac{y}{x^2} = \frac{-74}{14^2} =-0.378[/tex]. Thus a=0.378. Hence the correct expressions is y=-0.378*[tex]x^2[/tex]
2) Similarly, we have that the equation is [tex]x=3y^2 \\ \frac{1}{4p} =3[/tex]. Thus, p=1/12. In this case, the parabola opens along the x-axis and the focus is at (1/12, 0). Also, the directrix is at x=-1/12. Hence the correct answer is B.
3) We are given that the parabola has a p of 9. Also, the focus lies along the y-axis, hence the parabola is opening along the y-axis. Finally, the focus is on the positive half, so the parabola is opening upwards. The equation for this case is y=[tex] y=\frac{1}{4p} x^2= \frac{1}{36 } x^2[/tex].
4) Similarly as above. The directrix is superfluous, we only need the p-value. THe same comments about the parabola apply and if we substitute p=8 in the formula: [tex]y= \frac{1}{4p} x^2 [/tex] we get y=[tex] \frac{1}{32} x^2[/tex].
5) This is somewhat different, even though we do not need the directrix again. The focus lies on the x-axis, thus the parabola opens in this direction. The focus lies on the positive part of the axis, thus the parabola opens to the right. We also are given p=7. Hence, the equation we need is of the form[tex]x= \frac{1}{4p} y^2[/tex]. Substituting p=7, we get [tex]x= \frac{1}{28} y^2[/tex].
6) The equation of a prabola with a vertex at (0,0) is of the form y=-ax^2. The minus sign is needed since the parabola is downwards. Since we are given anothe point, we can calculate a. We have to take y=-74 and x=14 feet (since left to right is 28, we need to take half). [tex]-a= \frac{y}{x^2} = \frac{-74}{14^2} =-0.378[/tex]. Thus a=0.378. Hence the correct expressions is y=-0.378*[tex]x^2[/tex]
Answer:
1.
Given the parabolic equation:
[tex]x^2=20y[/tex]
The equation of parabola is given by:
[tex](x-h)^2 =4p(y-k)[/tex] .....[A]
where,
|4p| represents the focal width of the parabola
Focus = (h, k+p)
Vertex = (h, k)
Directrix (y) = k -p
On comparing given equation with equation [A] we have;
we have;
4p = 20
Divide both sides by 4 we have;
p = 5
Vertex =(0,0)
Focus = (0, 0+5) = (0, 5)
Focal width = 20
Directrix:
y = k-p = 0-5 = -5
⇒y = -5
Therefore, only option A is correct
2.
Given the parabolic equation:
[tex]x = 3y^2[/tex]
Divide both sides by 3 we have;
[tex]y^2 = \frac{1}{3}x[/tex]
The equation of parabola is given by:
[tex](y-k)^2 =4p(x-h)[/tex] ....[B]
Vertex = (h, k)
Focus = (h+p, k)
directrix: x = k -p
Focal width = 4p
Comparing given equation with equation [B] we have;
[tex]4p = \frac{1}{3}[/tex]
Divide both sides by 4 we have;
[tex]p = \frac{1}{12}[/tex]
Focal width = [tex]\frac{1}{3} = 0.33..[/tex]
Vertex = (0, 0)
Focus = [tex](0+\frac{1}{12}, 0) =(\frac{1}{12}, 0)[/tex]
directrix:
[tex]x = 0-\frac{1}{12}=-\frac{1}{12}[/tex]
⇒[tex]x = -\frac{1}{12}[/tex]
Therefore, option B is correct.
3.
The equation of parabola that opens upward is:
[tex]x^2 = 4py[/tex]
For the given problem:
Axis of symmetry:
x = 0
Distance from a focus to the vertex on the axis of the symmetry:
p = 9
then;
4p = 36
⇒[tex]x^2 = 36y[/tex]
Divide both sides by 36 we have;
[tex]y = \frac{1}{36}x^2[/tex]
Therefore, the only option A is correct.
4.
The equation of parabola that opens upward is:
[tex]x^2 = 4py[/tex]
Given that: Focus = (0, 8) and directrix: y = -8
Distance from a focus to the vertex and vertex to directrix is same:
i,e
|p| = 8
Then,
4p = 32
⇒[tex]x^2 = 32y[/tex]
Divide both sides by 32 we have;
[tex]y = \frac{1}{32}x^2[/tex]
Therefore, the only option A is correct.
5.
The equation of parabola that opens right is:
[tex]y^2 = 4px[/tex]
Given that:
Focus: (7, 0) and directrix: x = -7
Distance from a focus to the vertex and vertex to directrix is same:
i,e
|p| = 7
then
4p = 28
⇒[tex]y^2 =28x[/tex]
Divide both sides by 32 we have;
[tex]x= \frac{1}{28}y^2[/tex]
Therefore, the option B is correct.
6.
As per the statement:
A building has an entry the shape of a parabolic arch 74 ft high and 28 ft wide at the base as shown below.
The equation of parabola is given by:
[tex]x^2 = -4py[/tex] .....[C]
Substitute the point (14, -74) we have;
Put x = 14 and y = -74
then;
[tex](14)^2 = -4p \cdot (-74)[/tex]
⇒[tex]196 = 4p \cdot 74[/tex]
Divide both sides by 74 we have;
[tex]4p = \frac{196}{74} = \frac{98}{37}[/tex]
Substitute in the equation [C] we have;
[tex]x^2 = -\frac{98}{37}y[/tex]
or
[tex]y = -\frac{37}{98}x^2[/tex]
Therefore, an equation for the parabola if the vertex is put at the origin of the coordinate system is, [tex]y = -\frac{37}{98}x^2[/tex]
