Free Printable WorksheetMathPre-CalculusMath CurriculumHSG-GPE.A.1-3(+)

Pre CAL.121 Find Intersections of Lines and Conics Worksheet

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Worksheet Overview & Skills Covered

Core academic competencies and curriculum alignment

This Pre-Calculus Math worksheet provides structured practice on Pre CAL.121 Find Intersections of Lines and Conics. Students are guided through targeted exercises that build foundational understanding, computational accuracy, and critical thinking skills.

Pre-CAL.121 Find Intersections of Lines and Conics - Primary standard selected for prompt fields in this prompt cell: CCSS HSG-GPE.A.1-3(+). Skill statement in this prompt cell: Given a line and a conic, determine the intersection point(s) and select the correct set of solutions after checking validity. Associated standards listed with this skill in this prompt cell: CCSS HSG-GPE.A.1-3(+); CCSS HS-Modeling; SMP 1,2,4,5,7,8; AP Precalculus CED Units 3-4; TX §111.42(c)(3)(D-H); FL MA.912.GR.7.1-7.9, MA.912.T.4.5-4.7; VA MA.AG.1-2. Use this prompt-provided wording as the supporting standard description for this workflow.

Whether you are an educator preparing classroom handouts, a tutor reinforcing weekly topics, or a parent supporting homeschool learning, this printable resource gives students the deliberate practice required to achieve subject fluency and master state standards.

Grade

Pre-Calculus

Subject

Math

Domain

Math Curriculum

Formats

PDF & DOCX

Sample Practice Problems

Preview the types of problems and questions covered in this worksheet

Below are sample questions representative of the exercises included on this worksheet and related Pre CAL.121 Find Intersections of Lines and Conics skill assessments:

Problem 1Difficulty: Mastery

Solve this conic-and-line system exactly. Line: x - y = 2. Conic: x²/4 - y²/4 = 1. Check whether substituting produces a contradiction or valid points.

A{(0, 2)}

B{(2, 0)}

C{(-2, 0)}

D{(3, 0)}

Solution & Explanation:Substitute the line into the conic to get x - 2 = 0. Solving gives x = 2; substituting back into the line and checking both equations gives {(2, 0)}.

Problem 2Difficulty: Mastery

Which option gives the valid intersection points, with no extras? Line: x + y = 6. Conic: x²/64 + y²/16 = 1. Verify real roots before selecting the coordinate set.

A{(-4*sqrt(11)/5 + 6/5, 4*sqrt(11)/5 + 24/5), (6/5 + 4*sqrt(11)/5, -4*sqrt(11)/5 + 24/5)}

B{(-4*sqrt(11)/5 + 24/5, -4*sqrt(11)/5 - 6/5), (4*sqrt(11)/5 + 24/5, -6/5 + 4*sqrt(11)/5)}

C{(-4*sqrt(11)/5 + 24/5, 6/5 + 4*sqrt(11)/5), (4*sqrt(11)/5 + 24/5, -4*sqrt(11)/5 + 6/5)}

D{(-24/5 - 4*sqrt(11)/5, -4*sqrt(11)/5 + 6/5), (-24/5 + 4*sqrt(11)/5, 6/5 + 4*sqrt(11)/5)}

Solution & Explanation:Substitute the line into the conic to get 5*x² - 48*x + 80 = 0. Solving gives x = -4*sqrt(11)/5 + 24/5, 4*sqrt(11)/5 + 24/5; substituting back into the line and checking both equations gives {(-4*sqrt(11)/5 + 24/5, 6/5 + 4*sqrt(11)/5), (4*sqrt(11)/5 + 24/5, -4*sqrt(11)/5 + 6/5)}.

Problem 3Difficulty: Mastery

Find the real coordinate pair(s) shared by these two equations. Line: 2x + y = -3. Conic: x² + (y + 3)² = 9 (equivalent to r = -6sin(theta)). Answer in rectangular coordinates.

A{(-3 - 6*sqrt(5)/5, 3*sqrt(5)/5), (-3 + 6*sqrt(5)/5, -3*sqrt(5)/5)}

B{(-3*sqrt(5)/5, -6*sqrt(5)/5 + 3), (3*sqrt(5)/5, 6*sqrt(5)/5 + 3)}

C{(-3*sqrt(5)/5, -3 - 6*sqrt(5)/5), (3*sqrt(5)/5, -3 + 6*sqrt(5)/5)}

D{(-3*sqrt(5)/5, -3 + 6*sqrt(5)/5), (3*sqrt(5)/5, -3 - 6*sqrt(5)/5)}

Solution & Explanation:Substitute the line into the conic to get 5*x² - 9 = 0. Solving gives x = -3*sqrt(5)/5, 3*sqrt(5)/5; substituting back into the line and checking both equations gives {(-3*sqrt(5)/5, -3 + 6*sqrt(5)/5), (3*sqrt(5)/5, -3 - 6*sqrt(5)/5)}.

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Frequently Asked Questions

Common questions about printing, using, and solving this worksheet

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QWhat grade level and standards does this worksheet align with?

This worksheet is designed specifically for Pre-Calculus Math students. It aligns with educational standard HSG-GPE.A.1-3(+) (Primary standard selected for prompt fields in this prompt cell: CCSS HSG-GPE.A.1-3(+). Skill statement in this prompt cell: Given a line and a conic, determine the intersection point(s) and select the correct set of solutions after checking validity. Associated standards listed with this skill in this prompt cell: CCSS HSG-GPE.A.1-3(+); CCSS HS-Modeling; SMP 1,2,4,5,7,8; AP Precalculus CED Units 3-4; TX §111.42(c)(3)(D-H); FL MA.912.GR.7.1-7.9, MA.912.T.4.5-4.7; VA MA.AG.1-2. Use this prompt-provided wording as the supporting standard description for this workflow.).