# Honors Algebra 2: Extra Practice Conics

• Class: Honors Algebra 2
• Author: Peter Atlas
• Algebra and Trigonometry: Structure and Method, Brown

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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

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

circle

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

circle

7. $$\displaystyle 7x^2 + 7x + 7y - 5 = 7x^2$$
8. Solution

line

9. $$\displaystyle 2x^2 + 3x + 4y^2 + 5y - 10 = 0$$
10. Solution

ellipse

11. $$\displaystyle \frac{x-3}{4}+\frac{y + 1}{10} = 1$$
12. Solution

line

13. $$\displaystyle \frac{\left(x + 10\right)^2}{2\sqrt{3}}+\frac{\left(y-6\right)}{2\sqrt{3}} = 1$$
14. 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

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

4. 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.
5. 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 $$\displaystyle \left(-1, 2 - \sqrt{5} \right)$$

6. 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.
7. 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: $$x = -13$$

8. Write an equation for the parabola whose directrix is the line $$x = 2$$ and whose focus is at $$(1, 0)$$.
9. Solution

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

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

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

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

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