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Algebra II – Conic Sections

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    Study Guide Algebra II – Conic Sections 1. The Four Conic Sections Introduction Imagine cutting through a cone with a flat surface. The shape that appears where the cut happens is called a conic section . Conic sections are formed when a plane (a flat surface) slices through a pair of right circular cones that are placed tip to tip. The shape that forms depends on the angle at which the plane cuts through the cone . By changing this angle, different curves can be created. The four main conic sections are: • Circle • Parabola • Ellipse • Hyperbola These shapes are important in mathematics because they can be described using quadratic equations , 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 the cone meet is called a conic section . Each conic section has its own unique shape and mathematical properties. The four types of conic sections are: 1. Circle 2. Parabola 3. Ellipse

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    Study Guide 4. Hyperbola Figure 1. Creating conic sections. How Conic Sections Are Formed The type of conic section created depends entirely on how the plane slices through the cone . 1. Circle A circle is formed when the cutting plane is horizontal , meaning it is parallel to the base of the cone. The intersection creates a perfectly round shape. 2. Parabola A parabola is formed when the plane cuts through one cone at an angle , but not steep enough to intersect the second cone . This produces a curved shape that opens outward. Parabolas are commonly seen in satellite dishes, car headlights, and projectile motion . 3. Ellipse An ellipse forms when the plane cuts through one cone at a slanted angle , but the plane is not parallel to the side of the cone .

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    Study Guide The result is an oval - shaped curve . Ellipses are important in astronomy because planets move around the sun in elliptical orbits . 4. Hyperbola A hyperbola occurs when the plane cuts through both cones . This creates two separate curved branches that open away from each other. Hyperbolas appear in areas such as navigation systems and radio signal location . 2. Quiz: The Four Conic Sections 1. Question Forming conic sections requires intersection of ______. Answer Choices • a plane and 1 right 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 apexes Correct Answer : a plane and 2 stacked right circular cones connected at their apexes Why This Is Correct Conic sections are formed when a flat plane cuts through a double cone (two identical cones placed tip - to - tip at their apexes). This shape is called a double right circular cone . Depending on the angle of the plane cutting the double cone, different curves are produced such as circles, ellipses, parabolas, and hyperbolas. 2. Question

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    Study Guide A plane perpendicular to the axis of a single cone intersects it to form which of these? Answer Choices • a parabola • an ellipse • a circle Correct Answer : a circle Why This Is Correct When a plane cuts a cone perpendicular to the axis (straight across the cone), the intersection forms a circle . This happens because the slice is parallel to the circular base of the cone, producing a perfectly round cross - section. 3. Question A plane parallel to the axis of a double cone intersects both to form which of these? Answer Choices • an ellipse • a hyperbola • a parabola Correct Answer : a hyperbola Why This Is Correct When the cutting plane is parallel to the axis of the double cone , it passes through both cones (the upper and lower parts). This creates two separate curved branches, which together form a hyperbola .

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    Study Guide 4. Question A plane neither perpendicular nor parallel to the axis of a single cone intersects it to form which of these? Answer Choices • an ellipse • a circle • a parabola Correct Answer : an ellipse Why This Is Correct If the plane cuts the cone at an angle that is not perpendicular or parallel to the axis , and the plane does not pass through both cones , the result is an ellipse . A circle is actually a special case of an ellipse, but when the slice is tilted, the circular shape stretches into an oval curve called an ellipse. 5. Question A plane neither perpendicular nor parallel to the axis of a single cone intersects it to form which of these? Answer Choices • a circle • a parabola • a hyperbola Correct Answer : a parabola Why This Is Correct When the plane cuts the cone at an angle that is parallel to the slanted side (generator) of the cone , the intersection forms a parabola .

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    Study Guide The plane touches only one side of the cone and does not intersect the other half of the double cone, which creates the single open curve known as a parabola. 3. Circle What is a Circle? A circle is the set of all points in a plane that are the same distance from one fixed point. • The fixed point is called the center of the circle. • The distance from the center to any point on the circle is called the radius . 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 radius r , its equation is: This is called the standard form of the equation of a circle when the center is at the origin. Example 1 Graph the equation: x 2 + y 2 =16 Step 1: Identify the radius Compare the equation with the standard form: Here, So,

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    Study Guide Step 2: Identify the center Since the equation is in the form (x 2 + y 2 = r 2 ), the center is (0, 0) . Step 3: Key points on the circle Because the radius is 4, the circle passes through: • (4, 0) • ( - 4, 0) • (0, 4) • (0, - 4) These points help us draw the circle. Figure 1. Circle in standard position centered at (0,0) with radius 4. Example 2 Find the standard form of the equation of a circle centered at (0,0) with a radius of:

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    Study Guide Step 1: Use the standard formula For a circle centered at the origin: Step 2: Substitute the radius Square the radius: Step 3: Write the equation So, the equation of the circle is Equation of a Circle with Any Center If the center of the circle is (h, k) instead of (0,0), the standard form becomes: Where: • (h, k) = center of the circle • r = radius Special Case: Center at the Origin

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    Study Guide If the center is (0,0) , substitute (h=0) and (k=0): This simplifies to: Example 3 Graph the equation ( x – 3) 2 + ( y + 2) 2 = 25 Step 1: Compare with the standard form The standard form of a circle is: From the equation: • (h = 3) • (k = - 2) • (r 2 = 25) Step 2: Find the radius So, the circle has: • Center: (3, −2) • Radius: 5 This means the circle is shifted 3 units to the right and 2 units down from the origin. Step 3: Identify key points Starting from the center (3, −2), move 5 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 . Circle centered at (3, −2) with radius 5. The circle is shifted right and downward from the origin. Example 4 Find the standard form of the equation of the circle The circle has: • Center: (−6, 2) • Radius: ( 3√2 ) Step 1: Use the standard form formula

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