Hyperbolas and Conic Comparisons

Explore hyperbola definition, asymptotes and orientation, conic identification.

Lesson 4 of 43 parts Practice Flashcards

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Hyperbola Definition

Objective: Explain hyperbola definition and use it to solve or interpret a problem.

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.

Worked examples

Try itHyperbolas and Conic Comparisons — practice 1.1: For x²/16−y²/25=1, find focal distance squared.Show answer

Step 1: For a horizontal hyperbola, f²=A²+B². Step 2: Compute 16+25. Step 3: The squared focal distance is 41.

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Hyperbola Definition

Objective: Explain hyperbola definition and use it to solve or interpret a problem.

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.

Worked examples

Try itHyperbolas and Conic Comparisons — practice 1.1: For x²/16−y²/25=1, find focal distance squared.Show answer

Step 1: For a horizontal hyperbola, f²=A²+B². Step 2: Compute 16+25. Step 3: The squared focal distance is 41.

Asymptotes and Orientation

Objective: Explain asymptotes and orientation and use it to solve or interpret a problem.

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.

Worked examples

Try itHyperbolas and Conic Comparisons — practice 2.1: For x²/4−y²/256=1, find positive asymptote slope.Show answer

Step 1: Horizontal hyperbola asymptotes have slopes ±B/A. Step 2: Here A=2, B=16. Step 3: The positive slope is 16/2=8.

Conic Identification

Objective: Explain conic identification and use it to solve or interpret a problem.

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.

Worked examples

Try itHyperbolas and Conic Comparisons — practice 3.1: How many branches does x²/9−y²/121=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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