Honors Algebra 2: Practice: Quadratic Systems

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Determine whether each of the following is a circle, ellipse, parabola, or hyperbola. For circles, give the coordinates of the center and the radius. For ellipses, give the orientation, the coordinates of the center, vertices, foci and endpoints of the minor axis, the lengths of the major and minor axes. For parabolas, give the orientation, the coordinates of the vertex and focus, and the equation for the directrix. For hyperbolas, give the orientation, the coordinates of the center, vertices, foci and endpoints of the transverse axis, the equations of the asymptotes.

  1. \( \displaystyle x^2 + y^2 - 6x - 4y - 12 = 0\)
    2. Solution

    Circle with center \((3, 2)\) and radius 5.

  2. \( \displaystyle y = \frac{1}{12}x^2 +1\)
    3. Solution

    Parabola opening up with vertex \((0, 1),\) focus \((0, 4),\) and directrix \(y = -2\)

  3. \( \displaystyle x = 3 + 2y - y^2 \)
    4. Solution

    Parabola opening left with vertex \((4, 1),\) focus \( \displaystyle \left(\frac{15}{4}, 1\right),\) directrix \( \displaystyle x = \frac{17}{4}.\)

  4. \( \displaystyle 25x^2 + 4y^2 = 100\)
    5. Solution

    Vertical ellipse with vertices \((0, \pm 5),\) center \((0, 0),\) endpoints of minor axis \(( \pm 2, 0),\) foci \((0, \pm \sqrt{21}),\) major axis 10 and minor axis 4.

  5. \( \displaystyle \frac{(x - 5)^2}{4} + \frac{(y - 3)^2}{1} = 1\)
    6. Solution

    Horizontal ellipse with vertices \((3, 3),\) and \((7, 3),\) center \((5, 3),\) endpoints of minor axis \((5, 2)\) and \((5, 4).\) Major axis = 4, minor axis = 2. Foci at \( (5 \pm \sqrt{3}, 3).\)

  6. \( \displaystyle 4x^2 - 9y^2 = 36\)
    7. Solution

    Horizontal hyperbola with center \((0, 0),\) vertices \((\pm3, 0),\) asymptotes \( \displaystyle y = \pm \frac{2}{3}x,\) foci: \( ( \pm \sqrt{13}, 0).\)

  7. \( \displaystyle x^2 + y^2 - 4x - 6y + 4 = 0\)
    8. Solution

    Circle with radius 3 and center \((2, 3)\)

  8. \( \displaystyle x^2 + 25y^2 - 6x - 100y + 84 = 0\)
    9. Solution

    Horizontal ellipse with major axis 10, minor axis 2, center at \((3, 2),\) vertices at \((8, 2)\) and \((-2, 2),\) endpoints of minor axis at \((3, 3)\) and \((3, 1),\) and foci at \( (3 \pm 2 \sqrt{6}, 2).\)

  9. \( \displaystyle 2x^2 - y - 4x + 5 = 0\)
    10. Solution

    Parabola opening up with vertex at \((1, 3).\) Focus \( \displaystyle \left(1, \frac{25}{8} \right).\) Directrix \( \displaystyle y = \frac{23}{8.}\)

  10. \( \displaystyle 4x^2 + 4y^2 - 8x + 4y = 3\)
    11. Solution

    Circle with center at \( \displaystyle \left(1, -\frac{1}{2} \right)\) and radius \(\sqrt{2}.\)

  11. \( \displaystyle x^2 - 4y^2 - 2x + 16y - 19 = 0\)
    12. Solution

    Horizontal hyperbola with center at \((1, 2),\) vertices at \((3, 2)\) and \((-1, 2),\) asymptotes at \( \displaystyle y - 2 = \pm \frac{1}{2}\left(x - 1\right);\) foci at \((1 \pm \sqrt{5}, 2).\)

  12. \( \displaystyle x^2 - 4y^2 - 4y - 4 = -4y^2 \)
    13. Solution

    Parabola opens up with vertex at \((0, -1),\) focus at the origin, and directrix \(y = -2\)

  13. \( \displaystyle 2x^2 + 2y^2 + 4x + 4y - 4 = 0\)
    14. Solution

    Circle with center at \((-1, -1)\) and radius 2.

  14. \( \displaystyle x^2 - y^2 - 4x - 6y + 4 = 0\)
    15. Solution

    Vertical hyperbola with center \((2, -3),\) vertices at \((2, 0)\) and \((2, -6),\) asymptotes at \(y + 3 = \pm (x - 2),\) and foci at \((2, -3 \pm 3\sqrt{2}).\)

  15. \( \displaystyle y^2 - 9x^2 - 6y - 18x - 9 = 0\)
    16. Solution

    Vertical hyperbola with center at \((-1, 3),\) vertices at \((-1, 6)\) and \((-1, 0),\) asymptotes \(y - 3 = \pm 3(x + 1),\) and foci \((-1, 3 \pm \sqrt{10}).\)

  16. \( \displaystyle 9x^2 + y^2 + 18x - 6y + 9 = 0\)
    17. Solution

    Vertical ellipse with center \((-1, 3),\) major axis 6, minor axis 2, vertices \((-1, 6)\) and \((-1, 0),\) endpoints of minor axis \((0, 3)\) and \((-2, 3),\) and foci \((-1, 3 \pm 2\sqrt{2}).\)

  17. \( \displaystyle x^2 - y^2 + y = (2 + y)(2 - y)\)
    18. Solution

    Parabola facing down, vertex at \((0, 4),\) focus at \( \displaystyle \left(0, \frac{15}{4} \right),\) directrix \( \displaystyle y = \frac{17}{4}.\)