Write the equation of each conic section, given the following characteristics: a) Write the equation of an ellipse with center at (3, 2) and horizonta

jernplate8 2021-01-06 Answered
Write the equation of each conic section, given the following characteristics:
a) Write the equation of an ellipse with center at (3, 2) and horizontal major axis with length 8. The minor axis is 6 units long.
b) Write the equation of a hyperbola with vertices at (3, 3) and (-3, 3). The foci are located at (4, 3) and (-4, 3).
c) Write the equation of a parabola with vertex at (-2, 4) and focus at (-4, 4)
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Answered 2021-01-07 Author has 108 answers
a) Concept used:
The standard form of the equation of an ellipse with center (h, k) and major axis parallel to the x-axis is given below:
(xh)2a2+(yk)2b2=1:a>b
The length of the major axis is 2 a and the length of the minor axis is 2 b.
Calculation
Consider the center of the ellipse as (3, 2), the length of the horizontal major axis is 8 and the length of minor axis is 6.
Since the length of the major axis is 8, then 2a=8,a=4.
Since the length of the minor axis is 6, then 2b=6,b=3.
Hence, the standard equation of the ellipse with center (3, 2) and a=4,b=3 is given below:
(x3)242+(y2)232=1
(x3)216+(y2)29=1
(b) Consider the vertices of the hyperbola as (3, 3) and (-3, 3). The foci of the hyperbola are (4, 3) and (-4, 3).
The y-coordinates of the vertices and foci are the same, so the transverse axis is parallel to the x-axis.
Thus, the equation of the hyperbola will have the form (xh)2a2(yk)2b2=1.
The center is halfway between the vertices (-3, 3) and (3, 3).
Now, Applying the midpoint formula to get center of the hyperbola as below:
(h,k)=(332,3+32)=(0,3)
The length of the transverse axis 2a, is bounded by the vertices.
Hence, the distance between x-coordinates is obtain as follows:
2a=|3(3)|
2a=|6|=6
a=3
a2=9
The coordinates of the foci are (h±c,k).
So, we have (hc,k)=(4,3)and(h+c,k)=(4,3).
Solve the above coordinates for c as follows:
(hc,k)=(4,3)
hc=4,k=3
0c=4
(h,k)=(0,3)
c=4
Now obtain the value of b2
using the relation c2=a2+b2 with a=3,c=4 as follows:
c2=a2+b2
42=32+b2
b2=169
b2=7
Hence, the standard equation of the hyperbola with center (0, 3) and transverse axis parallel to the x-axis is given below:
(x0)29(y3)27=1
(c) Consider the vertex of the parabola as (-2, 4) and the focus as (-4, 4).
Here, the y- coordinates of the vertex and focus are same. Hence, this is a regular horizontal parabola, where y part is square.
Here, take the difference between x coordinate of the focus and vertex as p=4+2=2.
Hence, the equation of the parabola with vertex (-2, 4) and focus (-4, 4) is given below:
(yk)2=4p(xh)

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