Algebra II – Conic Sections - Quick Guides

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Study GuideAlgebra II–Conic Sections1. The Four Conic SectionsIntroductionImagine cutting through a cone with a flat surface. The shape that appears where the cut happens iscalled aconic section.Conic sections are formed when aplane(a flat surface) slices through apair of right circular conesthat are placed tip to tip.The shape that forms depends onthe angle at which the plane cuts through the cone. Bychanging this angle, different curves can be created.The four main conic sections are:•Circle•Parabola•Ellipse•HyperbolaThese shapes are important in mathematics because they can be described usingquadraticequations, which are equations that include squared variables.What is a Conic Section?When a flat plane cuts through one or both cones, the curved shape formed where the plane and thecone meet is called aconic section.Each conic section has its own unique shape and mathematical properties.The four types of conic sections are:1.Circle2.Parabola3.Ellipse

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Study Guide4.HyperbolaFigure 1.Creating conic sections.How Conic Sections Are FormedThe type of conic section created depends entirely onhow the plane slices through the cone.1. CircleAcircleis formed when the cutting plane ishorizontal, meaning it is parallel to the base of the cone.The intersection creates a perfectly round shape.2. ParabolaAparabolais formed when the plane cuts throughone cone at an angle, butnot steep enough tointersect the second cone.This produces a curved shape that opens outward.Parabolas are commonly seen insatellite dishes, car headlights, and projectile motion.3. EllipseAnellipseforms when the plane cuts throughone cone at a slanted angle, but the plane isnotparallel to the side of the cone.

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Study GuideThe result is anoval-shaped curve.Ellipses are important in astronomy becauseplanets move around the sun in elliptical orbits.4. HyperbolaAhyperbolaoccurs when the plane cuts throughboth cones.This createstwo separate curved branchesthat open away from each other.Hyperbolas appear in areas such asnavigation systems and radio signal location.2. Quiz: The Four Conic Sections1.QuestionForming conic sections requires intersection of ______.Answer Choices•a plane and 1right circular cone•a plane and 1 or 2 stacked right circular cones connected at their apexes•a plane and 2 stacked right circular cones connected at their apexesCorrect Answer:a plane and 2 stacked right circular cones connected at theirapexesWhy This Is CorrectConic sections are formed when a flat plane cuts through adouble cone(two identical cones placedtip-to-tip at their apexes).This shape is called adouble right circular cone.Depending on the angle of the plane cutting the double cone, different curves are produced such ascircles, ellipses, parabolas, and hyperbolas.2.Question

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Study GuideA plane perpendicular to the axis of a single cone intersects it to form which of these?Answer Choices•a parabola•an ellipse•a circleCorrect Answer:a circleWhy This Is CorrectWhen a plane cuts a coneperpendicular to the axis(straight across the cone), the intersectionforms acircle.This happens because the slice is parallel to the circular base of the cone, producing aperfectlyround cross-section.3.QuestionA plane parallel to the axis of a double cone intersects both to form which of these?Answer Choices•an ellipse•a hyperbola•a parabolaCorrect Answer:a hyperbolaWhy This Is CorrectWhen the cutting plane isparallel to the axis of the double cone, it passes throughboth cones(the upper and lower parts).This creates two separate curved branches, which together form ahyperbola.

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Study Guide4.QuestionA plane neither perpendicular nor parallel to the axis of a single cone intersects it to form which ofthese?Answer Choices•an ellipse•a circle•a parabolaCorrect Answer:an ellipseWhy This Is CorrectIf the plane cuts the cone at anangle that is not perpendicular or parallel to theaxis, and theplane doesnot pass through both cones, the result is anellipse.A circle is actually a special case of an ellipse, but when the slice is tilted, the circular shape stretchesinto an oval curve called an ellipse.5.QuestionA planeneither perpendicular nor parallel to the axis of a single cone intersects it to form which ofthese?Answer Choices•a circle•a parabola•a hyperbolaCorrect Answer:a parabolaWhy This Is CorrectWhen the plane cuts the cone at an angle that isparallel to the slanted side (generator) of thecone, the intersection forms aparabola.

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Study GuideThe plane touches onlyone side of the coneand does not intersect the other half of the doublecone, which creates the single open curve known as a parabola.3. CircleWhat is a Circle?Acircleis the set of all points in a plane that are the same distance from one fixed point.•The fixed point is called thecenterof the circle.•The distance from the center to any point on the circle is called theradius.Every point on the circle is exactly the same distance from the center.Equation of a Circle (Center at the Origin)If a circle has its center at the point(0, 0)and has a radiusr, its equation is:This is called thestandard form of the equation of a circlewhen the center is at the origin.Example 1Graph the equation:x2+y2=16Step 1:Identify the radiusCompare the equation with the standard form:Here,So,

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Study GuideStep 2:Identify the centerSince the equation is in the form (x2+ y2= r2), thecenter is (0, 0).Step 3:Key points on the circleBecause the radius is 4, the circle passes through:•(4, 0)•(-4, 0)•(0, 4)•(0,-4)These points help us draw thecircle.Figure 1.Circle in standard position centered at (0,0) with radius 4.Example 2Find thestandard form of the equation of a circlecentered at(0,0)with a radius of:

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Study GuideStep 1:Use the standard formulaFor a circle centeredat the origin:Step 2:Substitute the radiusSquare the radius:Step 3:Write the equationSo, theequation of the circle isEquation of a Circle with Any CenterIf the center of the circle is(h, k)instead of (0,0), the standard form becomes:Where:•(h, k)= center of the circle•r= radiusSpecial Case: Center at the Origin

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Study GuideIf the center is(0,0), substitute (h=0) and (k=0):This simplifies to:Example 3Graph the equation(x–3)2+ (y+ 2)2= 25Step 1:Compare with the standard formThe standard form of a circle is:From the equation:•(h = 3)•(k =-2)•(r2= 25)Step 2:Find the radiusSo, the circle has:•Center:(3, −2)•Radius:5This means the circle isshifted 3 units to the right and 2units downfrom the origin.Step 3:Identify key pointsStarting from the center (3, −2), move5 units in each direction:

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Study Guide•Right →(8, −2)•Left →(−2, −2)•Up →(3, 3)•Down →(3, −7)These points help us draw the circle.Figure 2.Circlecentered at (3, −2) with radius 5. The circle is shifted right and downward from theorigin.Example 4Find the standard form of the equation of the circleThe circle has:•Center:(−6, 2)•Radius:(3√2)Step 1:Use the standard form formula

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