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NY Regents Geometry Conic Sections in General Form — practice questions

Explore 6 free Conic Sections in General Form questions in NY Regents Geometry Mathematics. Open each question or exercise to read its content, reference answers and available explanation.

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Conic Sections in General Form questions and exercises with answers

Showing questions 1–6 of 6.

Which four curves are grouped together as the nondegenerate conic sections?Read the answer and worked solution →A plane that passes through the apex of the double cone produces which kind of result?Read the answer and worked solution →In the general form of a conic with no xy term, A equals 5 and C equals 5. Which curve is it?Read the answer and worked solution →For 4x squared plus 6xy plus 2y squared minus 7 equals 0, what does the discriminant show?Read the answer and worked solution →What stays unchanged when the axes are rotated to remove the xy term?Read the answer and worked solution →Sort the ellipse and hyperbola partsRead the answer and worked solution →

Worked practice questions

Which four curves are grouped together as the nondegenerate conic sections?

  1. Ellipse, point, hyperbola, line
  2. Circle, line, parabola, point
  3. Ellipse, circle, hyperbola, parabola
  4. Point, line, ray, hyperbola
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Correct answer: C. Ellipse, circle, hyperbola, parabola

1. Slicing a double cone with a plane that misses the apex gives a genuine curve. 2. Those four curves are the ellipse, the circle, the hyperbola and the parabola. 3. A point, a line and a pair of crossing lines are the degenerate cases instead. _Source: OpenStax College Algebra (CC BY 4.0), Ch 8 "Analytic Geometry", section 8.4 Rotation of Axes_

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A plane that passes through the apex of the double cone produces which kind of result?

  1. An ellipse or a circle only
  2. A parabola in every case
  3. A hyperbola in every case
  4. A point, a line, or two lines
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Correct answer: D. A point, a line, or two lines

1. A cut that misses the apex gives one of the four ordinary conics. 2. A cut through the apex collapses the curve into a degenerate case. 3. The three degenerate results are a point, a single line, or two intersecting lines. 4. An ellipse cannot appear here because the apex forces the cross-section to pinch shut. _Source: OpenStax College Algebra (CC BY 4.0), Ch 8 "Analytic Geometry", section 8.4 Rotation of Axes_

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In the general form of a conic with no xy term, A equals 5 and C equals 5. Which curve is it?

  1. A circle
  2. An ellipse
  3. A parabola
  4. A hyperbola
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Correct answer: A. A circle

1. Write the equation as Ax squared plus Cy squared plus Dx plus Ey plus F equals 0. 2. Both A and C are nonzero and both equal 5, so they match in size and in sign. 3. Equal nonzero coefficients of the same sign make the graph a circle. 4. An ellipse needs A and C to share a sign but differ in size, which fails here. _Source: OpenStax College Algebra (CC BY 4.0), Ch 8 "Analytic Geometry", section 8.4 Rotation of Axes_

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More Mathematics topics

Law of CosinesThe EllipseThe ParabolaConic Sections in Polar CoordinatesLaw of SinesThe HyperbolaVectorsGraphs of Polar Equations