Ellipses

Develop focal definition, standard axes, focal distance.

Lesson 5 of 63 parts Practice Flashcards

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Focal Definition

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

An ellipse is a set of points with a constant sum of distances to two foci. A circle is a special ellipse with coincident foci. Yes, with its two foci at the center. 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

$PF₁+PF₂=2A$

Worked examples

Try itFor x²/81+y²/49=1, what is the constant sum of distances from a point on the ellipse to its foci?Show answer

Step 1: The major semiaxis is A=9. Step 2: The defining focal-distance sum is 2A. Step 3: The sum is 18.

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Focal Definition

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

Formulas

$PF₁+PF₂=2A$

Worked examples

Try itFor x²/81+y²/49=1, what is the constant sum of distances from a point on the ellipse to its foci?Show answer

Step 1: The major semiaxis is A=9. Step 2: The defining focal-distance sum is 2A. Step 3: The sum is 18.

Standard Axes

Objective: Explain standard axes, apply the rule, and justify relevant restrictions.

For x²/A²+y²/B²=1 with A>B>0, the major semiaxis is A on the x-axis; exchanging denominators changes orientation. The larger positive squared semiaxis. 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

$x²/A²+y²/B²=1, A>B>0$

Worked examples

Try itFor x²/81+y²/49=1, find the major semiaxis length.Show answer

Step 1: The larger denominator is 81. Step 2: It lies beneath x², so the major axis is horizontal. Step 3: The semiaxis length is √81=9.

Focal Distance

Objective: Explain focal distance, apply the rule, and justify relevant restrictions.

For major semiaxis A and minor B, focal distance f satisfies f²=A²−B²; it is less than A when B>0. No, because f²=A²−B²<A². 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 x²/81+y²/49=1, find focal distance squared.Show answer

Step 1: Use f²=A²−B² for an ellipse. Step 2: Compute 81−49. Step 3: The squared focal distance is 32.

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