what shapes can you obtain by taking a cross section of a cone?

a circle
an ellipse
a parabola
a hyperbola

Question

what shapes can you obtain by taking a cross section of a cone?

a circle
an ellipse
a parabola
a hyperbola

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "direct_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Cross Sections of 3D Shapes" ], "new_concepts": [ "Conic Sections" ], "current_concepts": [ "Cross Sections of 3D Shapes", "Conic Sections" ] } ## Analyze horizontal and angled cross sections $$ \begin{aligned} &\text{Horizontal plane parallel to base} \implies \text{circle} \ &\text{Angled plane cutting through all generators} \implies \text{ellipse} \end{aligned} $$ ## Analyze steeper angled cross sections $$ \begin{aligned} &\text{Plane parallel to a generator line} \implies \text{parabola} \ &\text{Plane parallel to the cone axis (cutting both nappes)} \implies \text{hyperbola} \end{aligned} $$ ## Determine all possible shapes $$ \text{Possible shapes} = \{\text{circle}, \text{ellipse}, \text{parabola}, \text{hyperbola}\} $$

a circle

an ellipse

a parabola

a hyperbola

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Geometry", "Cross Sections of 3D Shapes" ] }

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Question

Explanation:

Analyze horizontal and angled cross sections

Analyze steeper angled cross sections

Determine all possible shapes

Answer: