ChalkBee
Make

CCSS HSG-GPE.A.3 Worksheets

Algebra 2 · Expressing Geometric Properties with Equations

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. This is a Common Core (+) standard, i.e. advanced content beyond the college-and-career-ready core, taught in Algebra 2 / Precalculus.

Below are free printable worksheets that support HSG-GPE.A.3. Print them as PDFs, or generate a fresh one. Every math answer key is computed in code, so it is never wrong.

Teaching guide

What students learn

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. This is a Common Core (+) standard, i.e. advanced content beyond the college-and-career-ready core, taught in Algebra 2 / Precalculus.

Worked example
For the ellipse (x + 1)²/36 + (y + 2)²/4 = 1, state the centre, the vertices, the co-vertices, and the foci.
Answer: Centre (-1, -2); vertices (5, -2) and (-7, -2); co-vertices (-1, 0) and (-1, -4); foci (-1 ± 42\sqrt{2}, -2)
Why: Compare with (x - h)²/a² + (y - k)²/b² = 1: centre (-1, -2). a² = 36 and b² = 4, so the semi-axes are 6 (x-direction) and 2 (y-direction). The larger denominator (36) is under x, so the major axis is horizontal and the vertices are (5, -2) and (-7, -2); the co-vertices are (-1, 0) and (-1, -4). c² = |36 - 4| = 32, so c = 42\sqrt{2} and the foci (on the major axis) are (-1 ± 42\sqrt{2}, -2).

Generated and checked in code, so it is correct.

Algebra worksheets

All algebra worksheets →
Algebra 2 Algebra
Set A · PDF · answer key with Pro
Make a worksheet for HSG-GPE.A.3
More Algebra 2 standards: HSF-IF.A.3 · HSF-BF.A.2 · HSA-SSE.B.4 · HSA-REI.C.6 · HSA-REI.C.7 · HSA-REI.C.8 · HSN-VM.C.7 · HSN-VM.C.8
← Browse all Common Core math standards