parabolas

find the focus and directrix of the following parabola:

$x - 4)^2 = 16(y + 2)\

focus: \((?, \quad)$
directrix: $y = \quad$

Question

parabolas

find the focus and directrix of the following parabola:

$x - 4)^2 = 16(y + 2)\

focus: \((?, \quad)$
directrix: $y = \quad$

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "quick" ], "needs_drawing": true, "concepts_used": [ "Parabola Properties", "Focus of Parabola", "Directrix of Parabola" ], "new_concepts": [], "current_concepts": [ "Parabola Properties", "Focus of Parabola", "Directrix of Parabola" ] } ## Identify the vertex and focal parameter $$ \begin{aligned} &(x - h)^2 = 4p(y - k) \ &(x - 4)^2 = 16(y + 2) \ &(h, k) = (4, -2) \ &4p = 16 \implies p = 4 \end{aligned} $$ ## Determine the focus $$ \begin{aligned} &\text{Focus} = (h, k + p) \ &\text{Focus} = (4, -2 + 4) = (4, 2) \end{aligned} $$ ## Determine the directrix $$ \begin{aligned} &\text{Directrix: } y = k - p \ &y = -2 - 4 = -6 \end{aligned} $$ Find the focus and directrix of the following parabola: $$(x - 4)^2 = 16(y + 2)$$ Focus: (4, 2) Directrix: $y =$ -6 { "elements": [ { "type": "point", "params": [[4, -2]], "properties": { "name": "Vertex (4, -2)", "size": 4, "color": "#583C87", "withLabel": true } }, { "type": "point", "params": [[4, 2]], "properties": { "name": "Focus (4, 2)", "size": 4, "color": "#8C55F2", "withLabel": true } }, { "type": "line", "params": [[-10, -6], [10, -6]], "properties": { "name": "y = -6", "strokeColor": "#F2557F", "strokeWidth": 2, "withLabel": true } }, { "type": "functiongraph", "params": [{"js": "Math.pow(x - 4, 2) / 16 - 2", "latex": "\frac{(x-4)^2}{16} - 2"}, -10, 10], "properties": { "strokeColor": "#5583F2", "strokeWidth": 3 } } ], "timestamps": [0.5, 1.0, 1.5] } { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Geometry", "Parabola Properties" ] }

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Question

Explanation:

Identify the vertex and focal parameter

Determine the focus

Determine the directrix

Answer: