CCSS.Math.Content.HSG-GPE.A.3
The Standard
(+) Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant.
Common Core State Standards for Mathematics
Teacher's field guide
What This Standard Means
What Students Need to Do
- Students place the foci on a coordinate plane and write distance expressions from a general point (x, y). They use the given constant to form an equation, remove radicals, and rewrite it in standard form.
What Mastery Looks Like
- Given focal coordinates and a fixed sum or difference, a student produces the correct standard form equation and shows the algebra. The student uses the center, vertices, and axis direction to check that the equation fits the given information.
Common Misconceptions
- Students often use the full given distance constant as a instead of half of it. They may swap the relationships c² = a² − b² and c² = a² + b². They also lose signs when squaring radicals or assume every hyperbola opens left and right.
How to Assess It
- Exit ticket: A hyperbola has foci at (−5, 0) and (5, 0), with a focal distance difference of 6. Write the radical equation and derive its standard form, showing each squaring step.
Lesson moves
Ways to Teach It
Push two pins into grid paper, loop string around them, and trace an ellipse while keeping the string taut.
Write: Why does a fixed sum make a closed curve, while a fixed difference makes two branches?
Run a card sort matching focal coordinates, distance conditions, sketches, and standard form equations, then require one written derivation per set.
Give two receivers, a signal time difference, and its distance equivalent, then have students sketch possible source locations as a hyperbola.
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Printable HSG-GPE.A.3 Worksheet
A ready-to-print activity worksheet aligned to CCSS.Math.Content.HSG-GPE.A.3, with an answer key for the teacher on its own page. No account needed.
PDF, US Letter, prints cleanly in black and white.
Download the worksheetLearning progression
Where This Standard Sits
Before This Standard
If students are struggling here, check these first.
- CCSS.Math.Content.8.G.B.8
Deriving conic equations requires writing distances from a variable point to each focus using the coordinate distance formula.
- CCSS.Math.Content.HSA-REI.A.2
Deriving conic equations from focus distances requires setting up radical distance equations and squaring them without changing the solution set.
- CCSS.Math.Content.HSG-GPE.A.1
Circle equations practice distance-based loci and completing squares, which support deriving and recognizing ellipse and hyperbola equations.
Keep exploring
Related Standards
- CCSS.Math.Content.HSG-GPE.A.2
Derive the equation of a parabola given a focus and directrix.
- CCSS.Math.Content.HSG-SRT.D.10
(+) Prove the Laws of Sines and Cosines and use them to solve problems.
- CCSS.Math.Content.HSN-CN.C.9
(+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.
- CCSS.Math.Content.HSF-TF.C.9
(+) Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.
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