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Question 2049: These are the hardest questions, I cannot figure them out. Can you help me?
What is the vertex ot the parabola with the equation (y - 1)2 = 4x?
What is the focus of the parabola with the equation (x - 1)2 + 32= 8y?
Find the center of a circle with the equation x2 + y2 + 2x + 4y - 9 = 0.
Find the radius of a circle with the equation x2 + y2 - 6x - 12y + 36 = 0.
Find the center of an ellipse with the equation 9x2 + 16y2 - 18x + 64y = 71.
Find the foci of an ellipse with the equation 7(x - 2)2 + 3(y - 2)2 = 21.
Find the length of the major and minor axes of an ellipse with the equation 16x2 + 25y2 + 32x - 150y = 159.
Answer by khwang(438)   (Show Source):
You can put this solution on YOUR website! These are basic questions about conic and not hard at all.
Also,you should know how to type the square or power.
1.What is the vertex ot the parabola with the equation (y - 1)^2 = 4x ?
Sol: The equations as (y - k)^2 = c(x -h) or c(y - k) = (x -h)^2
are parabolas with vertex (h,k).
For given (y - 1)^2 = 4x = 4(x-0) , the vertex is (0,1)
2.What is the focus of the parabola with the equation (x - 1)^2 + 32= 8y?
Sol: As in (1), (x - 1)^2 + 32= 8y can be converted to
(x - 1)^2 = 8(y -4), so its vertex is (1,4).
3.Find the center of a circle with the equation x^2 + y^2 + 2x + 4y - 9 = 0.
Sol: Make complete square: the equation converts to
x^2 + 2x + 1 + y^2 + 4y + 4 = 9+1+4 = 14, or
(x+1)^2 + (y+2)^2 = 14. So, the center of the circle is (-1,-2)
and the radius is sqrt(14).
4. Find the radius of a circle with the equation x2 + y2 - 6x - 12y + 36 = 0.
Sol: As in (3):x^2 - 6x + (6/2)^2 + y2 - 12y + (12/2)^2 = -36 + 9 + 36 =9.
Or (x -3)^2 + (y-6)^2 = 9.
So, the center of the circle is (3,6)
and the radius is 3.
5.Find the center of an ellipse with the equation 9x^2 +16y^2 -18x+64y= 71.
Sol: Make complete square: the equation converts to
9(x^2 - 2x + 1) + 16(y^2 + 4y + 4) = 71 + 9 + 64 = 144, or
9(x-1)^2 + 16(y + 2)^2 = 144. Dividing each term by 144, we have:
(x-1)^2/16 + (y + 2)^2/9 = 1.
The center of the ellipse is (1,-2).
[also, the length of the major axis = 4 and the length of the minor axis
= 3]
6.Find the foci of an ellipse with the equation 7(x - 2)^2 + 3(y - 2)^2 = 21.
Sol: Dividing each term by 21, we have:
(x-1)^2/3 + (y - 2)^2/7 = 1.
We see the center is (1,2) and the length of the major axis a = sqrt(7),
the length of the minor axis b = sqrt(3). Note, the long axis is along
y-direction(vertical).
Since c^2 = a^2-b^2 = 7-3 =4, we get c = 2.
Hence, the foci are (1,2+c) = (1,4) and (1,2-c) = (1,0)
7.Find the length of the major and minor axes of an ellipse with the equation Sol: As in (5), the equation converts to
16(x^2 + 2x + 1) + 25(y^2 - 6y + 9) = 159 +16 +225 = 400 or
4(x+1)^2 + 25(y-3)^2 = 200.
Dividing each term by 200, we have:
(x+1)^2/50 + (y - 2)^2/8 = 1.
Hence, the major axis a = sqrtt(50) = 5 sqrt(2),
the minor axis b = sqrtt(8) = 2 sqrt(2).
Kenny
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