Hyperbolas and Conic Comparisons

Develop hyperbola definition, asymptotes and orientation, conic identification.

Lesson 6 of 63 parts Practice Flashcards

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Hyperbola Definition

Objective: Explain hyperbola definition, apply the rule, and justify relevant restrictions.

A hyperbola has a constant absolute difference of distances to two foci and two branches. It differs from an ellipse constant sum. It uses an absolute difference rather than a sum. Identify the stated domain before using the rule, preserve restrictions during algebraic simplification, and interpret the result in the units or graph features requested. Use the worked example to connect the symbolic rule with a concrete calculation.

Formulas

$c²=A²+B²$

Worked examples

Try itFor x²/9−y²/49=1, find focal distance squared.Show answer

Step 1: For a horizontal hyperbola, f²=A²+B². Step 2: Compute 9+49. Step 3: The squared focal distance is 58.

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Hyperbola Definition

Objective: Explain hyperbola definition, apply the rule, and justify relevant restrictions.

Formulas

$c²=A²+B²$

Worked examples

Try itFor x²/9−y²/49=1, find focal distance squared.Show answer

Step 1: For a horizontal hyperbola, f²=A²+B². Step 2: Compute 9+49. Step 3: The squared focal distance is 58.

Asymptotes and Orientation

Objective: Explain asymptotes and orientation, apply the rule, and justify relevant restrictions.

For x²/A²−y²/B²=1, branches open left/right and asymptotes have slopes ±B/A through the center. The positive term indicates transverse orientation. The branches open up and down. Identify the stated domain before using the rule, preserve restrictions during algebraic simplification, and interpret the result in the units or graph features requested. Use the worked example to connect the symbolic rule with a concrete calculation.

Formulas

$x²/A²−y²/B²=1 ⇒ y=±(B/A)x$

Worked examples

Try itFor x²/9−y²/441=1, find positive asymptote slope.Show answer

Step 1: Horizontal hyperbola asymptotes have slopes ±B/A. Step 2: Here A=3, B=21. Step 3: The positive slope is 21/3=7.

Conic Identification

Objective: Explain conic identification, apply the rule, and justify relevant restrictions.

A squared x plus squared y can represent ellipse/circle under suitable coefficients, whereas a difference of squares can represent a hyperbola. A single squared variable can represent a parabola. No, degeneracies and empty real sets are possible. Identify the stated domain before using the rule, preserve restrictions during algebraic simplification, and interpret the result in the units or graph features requested. Use the worked example to connect the symbolic rule with a concrete calculation.

Formulas

Horizontal hyperbola: x²/A²−y²/B²=1

Worked examples

Try itHow many branches does x²/9−y²/49=1 have?Show answer

Step 1: A difference of squared coordinate terms gives a hyperbola in this standard form. Step 2: A hyperbola has two disconnected branches. Step 3: Therefore the answer is 2.

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