Step-by-step explanation
The relations between the vertex, the focus and the directrix are given in the following graph:
A) F(4,6) and D: x = 0
In this case, we have:
0. D: x = 0 (vertical Directrix) → ,Case 2, and h - p = 0 → ,h = p,,
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1. F(4,6) → the focus is at the right of the Directrix → ,opens to the right, and ,h + p = 4, and ,k = 6,
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2. from equations of point 1 and 2, we have: 2h = 4 → ,h = 2,
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3. from point 2 and 3, we have V(h, k) = ,V(2,6),.
B) F(-2,5) and V(-2,1)
In this case, we have:
0. Focus and Vertex with x = -2 → Vertical axis of symmetry → Horizontal Directrix → ,Case 1,,
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1. The Focus is over the vertex → ,opens upward,,
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2. The distance between the Focus and the Vertex is ,p = 4,,
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3. The Vertex has coordinates V(-2,1) → (h, k) = (-2, 1),
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4. using points 3 and 4, we have y = k - p = 1 - 4 = -3 → the Directrix is ,D: y = -3,.
C) D: y=-4 and V(-2,1)
In this case, we have:
0. The Directrix is horizontal → ,Case 1,,
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1. the Directrix is below the Vertex → ,opens upward,,
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2. the distance between the Directrix and the Vertex is p = 1 - (-4) = 5,
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3. the Focus has coordinates ,F:, (h, k) = (-2, 5 + 1) → ,F(-2, 6),.
D) F(10, -8) and V(6,-8)
In this case, we have:
0. The axis of symmetry is y = -8 → is a horizontal axis → ,Case 2,,
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1. the focus is at the right of the vertex →, opens to the right,,
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2. the distance between the focus and the vertex is ,p = 10 - 6 = 4,,
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3. the directrix is at x = h - p = 6 - 4 = 2 → ,D: x = 2,.
Answers
A)
• Opens to the right
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• V(2,6)
B)
• Opens upward
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• D: y = -3
C)
• Opens upward
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• F(-2,6)
D)
• Opens to the right
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• D: x = 2