Math  /  Geometry

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Question Find the vertices of the ellipse defined by the equation shown below. If necessary, round to the nearest tenth 4x2+25y2+24x200y+336=04 x^{2}+25 y^{2}+24 x-200 y+336=0
Answer Attempt 2 out of 2
Vertices: \square , \square and \square , \square Submit Answer

Studdy Solution

STEP 1

1. The given equation represents an ellipse.
2. The equation needs to be rewritten in standard form to find the vertices.
3. The standard form of an ellipse is (xh)2a2+(yk)2b2=1\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1.

STEP 2

1. Rearrange and group terms for completing the square.
2. Complete the square for both xx and yy.
3. Rewrite the equation in standard form.
4. Identify the center and lengths of the semi-major and semi-minor axes.
5. Calculate the vertices.

STEP 3

Rearrange the equation by grouping xx and yy terms:
4x2+24x+25y2200y=336 4x^2 + 24x + 25y^2 - 200y = -336

STEP 4

Complete the square for the xx terms:
4(x2+6x) 4(x^2 + 6x)
To complete the square, add and subtract (62)2=9(\frac{6}{2})^2 = 9:
4(x2+6x+99)=4((x+3)29)=4(x+3)236 4(x^2 + 6x + 9 - 9) = 4((x+3)^2 - 9) = 4(x+3)^2 - 36
Complete the square for the yy terms:
25(y28y) 25(y^2 - 8y)
To complete the square, add and subtract (82)2=16(\frac{8}{2})^2 = 16:
25(y28y+1616)=25((y4)216)=25(y4)2400 25(y^2 - 8y + 16 - 16) = 25((y-4)^2 - 16) = 25(y-4)^2 - 400

STEP 5

Rewrite the equation using the completed squares:
4(x+3)236+25(y4)2400=336 4(x+3)^2 - 36 + 25(y-4)^2 - 400 = -336
Combine and simplify:
4(x+3)2+25(y4)2=100 4(x+3)^2 + 25(y-4)^2 = 100
Divide through by 100 to express in standard form:
(x+3)225+(y4)24=1 \frac{(x+3)^2}{25} + \frac{(y-4)^2}{4} = 1

STEP 6

Identify the center (h,k)(h, k), and lengths of semi-major axis aa and semi-minor axis bb:
Center: (3,4)(-3, 4)
Since a2=25a^2 = 25, a=5a = 5; and b2=4b^2 = 4, b=2b = 2.

STEP 7

Calculate the vertices:
The major axis is horizontal, so the vertices are at (3±5,4)(-3 \pm 5, 4):
Vertices: (8,4)(-8, 4) and (2,4)(2, 4)
The vertices of the ellipse are:
(8,4) \boxed{(-8, 4)} and (2,4) \boxed{(2, 4)}

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