parabolas

find the focus and directrix of the following parabola:

$x - 5)^2 = 8(y - 1)\

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

Question

parabolas

find the focus and directrix of the following parabola:

$x - 5)^2 = 8(y - 1)\

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": [], "new_concepts": [], "current_concepts": [ "Parabola Properties", "Focus of Parabola", "Directrix of Parabola" ] } ## Identify the vertex and focal parameter $$ \begin{aligned} &(x - h)^2 = 4p(y - k) \ &(x - 5)^2 = 8(y - 1) \ &(h, k) = (5, 1) \ &4p = 8 \implies p = 2 \end{aligned} $$ ## Calculate the focus coordinates $$ \begin{aligned} &\text{Focus} = (h, k + p) \ &\text{Focus} = (5, 1 + 2) = (5, 3) \end{aligned} $$ ## Determine the directrix equation $$ \begin{aligned} &\text{Directrix: } y = k - p \ &y = 1 - 2 \ &y = -1 \end{aligned} $$ Find the focus and directrix of the following parabola: $$(x - 5)^2 = 8(y - 1)$$ Focus: (5, 3) Directrix: $y =$ -1 { "elements": [ { "type": "point", "params": [[5, 1]], "properties": { "name": "Vertex (5,1)", "size": 4, "color": "#583C87", "withLabel": true } }, { "type": "point", "params": [[5, 3]], "properties": { "name": "Focus (5,3)", "size": 4, "color": "#8C55F2", "withLabel": true } }, { "type": "line", "params": [[0, -1], [10, -1]], "properties": { "name": "y = -1", "strokeColor": "#F2557F", "strokeWidth": 2, "withLabel": true, "labelPosition": [8, -0.5] } }, { "type": "functiongraph", "params": [{"js": "(x-5)*(x-5)/8 + 1", "latex": "\frac{(x-5)^2}{8} + 1"}, -2, 12], "properties": { "strokeColor": "#5583F2", "strokeWidth": 2 } } ] } { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Geometry", "Focus of Parabola" ] }

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QUESTION IMAGE

Question

Explanation:

Identify the vertex and focal parameter

Calculate the focus coordinates

Determine the directrix equation

Answer: