Conic Sections flashcards

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In Circles, what does center-radius form mean?Circles: An equation for points a fixed distance from a center.
Why take a positive square root for r?Circles: Radius is a nonnegative distance.
In Circles, what does completing the square mean?Circles: An algebraic rewrite from expanded circle equation.
Why balance added square terms?Circles: Adding to only one side changes the circle.
In Circles, what does coordinate distance mean?Circles: The straight-line distance between two plane points.
Can radius be negative?Circles: No, a distance cannot be negative.
In Parabolas as Loci, what does focus and directrix mean?Parabolas as Loci: A conic locus defined by equal distances.
Is the vertex halfway between focus and directrix?Parabolas as Loci: Yes, along the symmetry axis.
In Parabolas as Loci, what does vertical standard form mean?Parabolas as Loci: A focus-directrix equation for a vertical parabola.
What does negative p mean?Parabolas as Loci: The parabola opens downward.
In Parabolas as Loci, what does horizontal standard form mean?Parabolas as Loci: A focus-directrix equation for a horizontal parabola.
How does p affect its direction?Parabolas as Loci: Positive p opens right; negative p opens left.
In Ellipses, what does focal definition mean?Ellipses: A conic defined by a fixed sum of focal distances.
Is every circle also an ellipse?Ellipses: Yes, with its two foci at the center.
In Ellipses, what does standard axes mean?Ellipses: Major and minor semiaxes read from standard form.
Which denominator marks the major axis?Ellipses: The larger positive squared semiaxis.
In Ellipses, what does focal distance mean?Ellipses: The distance from center to each ellipse focus.
Could focal distance exceed the major semiaxis?Ellipses: No, because f²=A²−B²<A².
In Hyperbolas and Conic Comparisons, what does hyperbola definition mean?Hyperbolas and Conic Comparisons: A conic with two separate branches.
How does its focal condition differ from an ellipse?Hyperbolas and Conic Comparisons: It uses an absolute difference rather than a sum.
In Hyperbolas and Conic Comparisons, what does asymptotes and orientation mean?Hyperbolas and Conic Comparisons: Lines describing distant branch directions.
What changes if y² has the positive term?Hyperbolas and Conic Comparisons: The branches open up and down.
In Hyperbolas and Conic Comparisons, what does conic identification mean?Hyperbolas and Conic Comparisons: Classifying basic axis-aligned conic equations.
Can coefficients alone determine every graph without checking constants?Hyperbolas and Conic Comparisons: No, degeneracies and empty real sets are possible.