parabolas

find the focus and directrix of the following parabola:

$y - 4)^2 = 16(x - 6)\

focus: \((?, \quad)$

directrix: $x = \quad$

Question

parabolas

find the focus and directrix of the following parabola:

$y - 4)^2 = 16(x - 6)\

focus: \((?, \quad)$

directrix: $x = \quad$

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": [ "Parabola Properties", "Focus of Parabola", "Directrix of Parabola" ], "current_concepts": [ "Parabola Properties", "Focus of Parabola", "Directrix of Parabola" ] } ## Identify vertex and focal parameter $$ \begin{aligned} &(y - k)^2 = 4p(x - h) \ &(y - 4)^2 = 16(x - 6) \ &(h, k) = (6, 4) \ &4p = 16 \implies p = 4 \end{aligned} $$ ## Calculate the focus $$ \begin{aligned} &\text{Focus} = (h + p, k) \ &\text{Focus} = (6 + 4, 4) = (10, 4) \end{aligned} $$ ## Calculate the directrix $$ \begin{aligned} &\text{Directrix: } x = h - p \ &x = 6 - 4 = 2 \end{aligned} $$ Find the focus and directrix of the following parabola: $$(y - 4)^2 = 16(x - 6)$$ Focus: (10, 4) Directrix: $x =$ 2 { "elements": [ { "type": "point", "params": [[6, 4]], "properties": { "name": "V(6,4)", "size": 4, "color": "#8C55F2", "withLabel": true } }, { "type": "point", "params": [[10, 4]], "properties": { "name": "F(10,4)", "size": 4, "color": "#F2557F", "withLabel": true } }, { "type": "line", "params": [[2, -10], [2, 10]], "properties": { "name": "x=2", "strokeColor": "#5583F2", "strokeWidth": 2, "dash": 2, "withLabel": true } }, { "type": "curve", "params": [ { "x": { "js": "t*t/16 + 6", "latex": "y^2/16 + 6" }, "y": { "js": "t + 4", "latex": "y + 4" } }, -10, 10 ], "properties": { "strokeColor": "#8C55F2", "strokeWidth": 2 } } ], "timestamps": [0.5, 1.0, 1.5] } { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Geometry", "Focus of Parabola" ] }

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Question

Explanation:

Identify vertex and focal parameter

Calculate the focus

Calculate the directrix

Answer: