Lesson 2: Circles — Practice Questions

1. 1. Find the center and radius of the circle with the equation $(x - 2)^2 + (y + 3)^2 = 16$.

   • A. Center: (2, -3), Radius: 4
   • B. Center: (-2, 3), Radius: 16
   • C. Center: (2, -3), Radius: 16
   • D. Center: (-2, 3), Radius: 4

2. 2. Write the equation $x^2 + y^2 - 4x + 2y - 4 = 0$ in standard form.

   • A. $(x - 2)^2 + (y + 1)^2 = 9$
   • B. $(x + 2)^2 + (y - 1)^2 = 9$
   • C. $(x - 2)^2 + (y + 1)^2 = 4$
   • D. $(x - 4)^2 + (y + 2)^2 = 4$

3. 3. Find the solutions to the system of equations: $x^2 + y^2 = 17$ and $2xy = -17$.

   • A. $(\frac{\sqrt{34}}{2}, -\frac{\sqrt{34}}{2}), (-\frac{\sqrt{34}}{2}, \frac{\sqrt{34}}{2})$
   • B. $(\frac{\sqrt{17}}{2}, -\frac{\sqrt{17}}{2}), (-\frac{\sqrt{17}}{2}, \frac{\sqrt{17}}{2})$
   • C. $(\sqrt{17}, -1), (-\sqrt{17}, 1)$
   • D. No real solutions.

4. 4. Find the center of the circle given by the equation $x^2 + y^2 - 6y - 4 = 0$.

   • A. (0, 3)
   • B. (0, -3)
   • C. (0, 6)
   • D. (3, 0)

5. 5. Identify the conic section given by the equation $x^2 + y^2 = 9$ and its key feature.

   • A. A circle with radius 3
   • B. A circle with radius 9
   • C. An ellipse with major axis 3
   • D. A parabola with vertex at (0,0)

6. 6. To complete the square for the expression $x^2 - 12x$, what number must be added? \n\n___

7. 7. Which equation represents the standard form of the circle $x^2 + 2x + y^2 - 4y = 4$?

   • A. $(x+1)^2 + (y-2)^2 = 9$
   • B. $(x-1)^2 + (y+2)^2 = 9$
   • C. $(x+1)^2 + (y-2)^2 = 4$
   • D. $(x+2)^2 + (y-4)^2 = 4$

8. 8. The equation of a circle is $x^2 - 10x + y^2 + 8y = -1$. After converting to standard form $(x-h)^2 + (y-k)^2 = r^2$, what is the value of $k$? \n\n___

9. 9. When the equation $x^2 + 4x + y^2 - 14y = 4$ is written in the form $(x-h)^2 + (y-k)^2 = r^2$, what is the value of $r^2$? \n\n___

10. 10. What are the coordinates of the center of the circle with the equation $x^2 + y^2 + 6x - 12y = 19$?

    • A. (-3, 6)
    • B. (3, -6)
    • C. (-6, 12)
    • D. (6, -12)