parabolas

find the focus and directrix of the following parabola:

$y + 3)^2 = 8(x + 3)\

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

Question

parabolas

find the focus and directrix of the following parabola:

$y + 3)^2 = 8(x + 3)\

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 the standard form and vertex parameters $$ \begin{aligned} &(y - k)^2 = 4p(x - h) \ &(y + 3)^2 = 8(x + 3) \ &(h, k) = (-3, -3) \end{aligned} $$ ## Calculate the focal length parameter $$ \begin{aligned} &4p = 8 \ &p = 2 \end{aligned} $$ ## Determine the focus and directrix $$ \begin{aligned} &\text{Focus: } (h + p, k) = (-3 + 2, -3) = (-1, -3) \ &\text{Directrix: } x = h - p = -3 - 2 = -5 \end{aligned} $$ Find the focus and directrix of the following parabola: $$(y + 3)^2 = 8(x + 3)$$ Focus: (-1, -3) Directrix: $x =$ -5 { "elements": [ { "type": "point", "params": [[-3, -3]], "properties": { "name": "Vertex(-3,-3)", "size": 4, "color": "#583C87", "withLabel": true } }, { "type": "point", "params": [[-1, -3]], "properties": { "name": "Focus(-1,-3)", "size": 4, "color": "#8C55F2", "withLabel": true } }, { "type": "line", "params": [[-5, 0], [-5, 1]], "properties": { "name": "x = -5", "strokeColor": "#F2557F", "strokeWidth": 2, "withLabel": true } }, { "type": "functiongraph", "params": [ { "js": "Math.sqrt(8*(x+3)) - 3", "latex": "\sqrt{8(x+3)} - 3" }, -3, 10 ], "properties": { "strokeColor": "#5583F2", "strokeWidth": 2 } }, { "type": "functiongraph", "params": [ { "js": "-Math.sqrt(8*(x+3)) - 3", "latex": "-\sqrt{8(x+3)} - 3" }, -3, 10 ], "properties": { "strokeColor": "#5583F2", "strokeWidth": 2 } } ], "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 IMAGE

Question

Explanation:

Identify the standard form and vertex parameters

Calculate the focal length parameter

Determine the focus and directrix

Answer: