Honors Algebra 2: Extra Practice Conics

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  1. Identify the following by what kind of conic section it is:
    1. \( \displaystyle x = -3\left(y + 1\right)^2 + 2\)
      2. Solution

      sideways parabola

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

      circle

    3. \( \displaystyle 5x^2 + 3x + 5y^2 + 4y - 7 = 0\)
      4. Solution

      circle

    4. \( \displaystyle 7x^2 + 7x + 7y - 5 = 7x^2\)
      5. Solution

      line

    5. \( \displaystyle 2x^2 + 3x + 4y^2 + 5y - 10 = 0\)
      6. Solution

      ellipse

    6. \( \displaystyle \frac{x-3}{4}+\frac{y + 1}{10} = 1\)
      7. Solution

      line

    7. \( \displaystyle \frac{\left(x + 10\right)^2}{2\sqrt{3}}+\frac{\left(y-6\right)}{2\sqrt{3}} = 1\)
      8. Solution

      parabola

  2. Write an equation for the ellipse whose foci are at \((5, -1)\) and \( (-1, -1) \), and the sum of whose focal radii is 8.
    3. Solution

    \( \displaystyle \frac{\left(x - 2\right)^2}{16} + \frac{\left(y +1\right)^2}{7} = 1 \)

  3. For the ellipse: \( \displaystyle \frac{\left(x + 1 \right)^2} {4}+\frac{\left(y - 2\right)^2}{9} = 1,\) list the orientation, major axis, minor axis, center, vertices, endpoints of minor axis, and foci.
    4. Solution

    vertical; major axis: 6; minor axis: 4; Center \((-1, 2);\) Vertices: \((-1, 5)\) and \((-1, -1);\) Endpoints of minor axis: \((-3, 2)\) and \((1, 2);\) Foci: \( \displaystyle \left(-1, 2 + \sqrt{5}\right) \) and \( \displaystyle \left(-1, 2 - \sqrt{5} \right) \)

  4. For the parabola: \( \displaystyle x = \frac{1}{2}y^2 + y - 12,\) list the orientation, the vertex, the x-intercept(s), the y-intercept(s), the axis of symmetry, the coordinates of the point (6, 12) reflected across the axis of symmetry, the focus, and the directrix.
    5. Solution

    Opens right; Vertex: \( \displaystyle \left(- \frac{25}{2}, -1 \right);\) x-intercept: \( (-12, 0); \) y-intercepts: \((0, -6)\) and \((0, 4);\) Axis of symmetry: y = -1; Reflection: (-8, 12); Focus: \((-12, -1);\) Directrix: \(x = -13\)

  5. Write an equation for the parabola whose directrix is the line \( x = 2\) and whose focus is at \((1, 0)\).
    6. Solution

    \( \displaystyle x = -\frac{1}{2}y^2 + \frac{3}{2} \)

  6. Write the equation for a circle whose center is (4, 3) and that is tangent to the y-axis
    7. Solution

    \( \displaystyle \left(x - 4\right)^2 + \left(y - 3\right)^2 = 16\)

  7. Find the center, orientation, lengths of the major and transverse axes, and foci of the ellipse whose equation is \( \displaystyle 9x^2 + 4y^2 + 18x - 16y - 11 = 0\)
    8. Solution

    Center: \((-1, 2);\) Orientation: vertical; Major axis: 6; Minor axis: 4; Foci: \( \displaystyle \left(-1, 2 - \sqrt{5}\right)\) and \( \displaystyle \left(-1 + 2 + \sqrt{5}\right)\)