find the focus and directrix of the following parabola:

$y + 3)^2 = 16(x - 4)\

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

Question

find the focus and directrix of the following parabola:

$y + 3)^2 = 16(x - 4)\

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": [], "current_concepts": [ "Parabola Properties", "Focus of Parabola", "Directrix of Parabola" ] } ## Identify the vertex and focal parameter $$ \begin{aligned} &(y - k)^2 = 4p(x - h) \ &(y + 3)^2 = 16(x - 4) \ &(h, k) = (4, -3) \ &4p = 16 \implies p = 4 \end{aligned} $$ ## Calculate the focus coordinates $$ \begin{aligned} &\text{Focus} = (h + p, k) \ &\text{Focus} = (4 + 4, -3) = (8, -3) \end{aligned} $$ ## Determine the directrix equation $$ \begin{aligned} &\text{Directrix: } x = h - p \ &x = 4 - 4 \ &x = 0 \end{aligned} $$ Find the focus and directrix of the following parabola: $$(y + 3)^2 = 16(x - 4)$$ Focus: (8, -3) Directrix: $x =$ 0 { "elements": [ { "type": "point", "params": [[4, -3]], "properties": { "name": "Vertex (4, -3)", "size": 4, "color": "#583C87", "withLabel": true } }, { "type": "point", "params": [[8, -3]], "properties": { "name": "Focus (8, -3)", "size": 4, "color": "#8C55F2", "withLabel": true } }, { "type": "line", "params": [[0, -10], [0, 10]], "properties": { "name": "Directrix: x = 0", "strokeColor": "#F2557F", "strokeWidth": 2, "dash": 2, "withLabel": true } }, { "type": "curve", "params": [ { "x": { "js": "t*t/16 + 4", "latex": "y^2/16 + 4" }, "y": { "js": "t - 3", "latex": "y" } }, -10, 10 ], "properties": { "strokeColor": "#5583F2", "strokeWidth": 2 } } ] } { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Geometry", "Parabola Properties" ] }

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QUESTION IMAGE

Question

Explanation:

Identify the vertex and focal parameter

Calculate the focus coordinates

Determine the directrix equation

Answer: