Derive the Equation of a Circle - Geometry

1. What is the standard form of $x^2+y^2+2x-8y+5=0$?

   $(x+1)^2+(y-4)^2=12$. From general form after completing the square.

2. What is the rule for the constant used to complete the square for $u^2+bu$?

   Add $\left(\frac{b}{2}\right)^2$. Square half the linear coefficient.

3. What are the center and radius of $x^2+y^2+8x+2y-8=0$?

   Center $(-4,-1)$, radius $5$. Center from $(x+4)^2+(y+1)^2=25$, $r=5$.

4. What condition on $r^2$ in $(x-h)^2+(y-k)^2=r^2$ gives no real circle?

   $r^2<0$. Negative right side has no real solutions.

5. What is the standard form after completing the square for $x^2+y^2-4x+14y+29=0$?

   $(x-2)^2+(y+7)^2=24$. Complete square: $(x-2)^2-4+(y+7)^2-49+29=0$.

6. What is the center and radius of the circle $(x+2)^2+(y-1)^2=16$?

   Center $(-2,1)$, radius $4$. From $(x+2)^2=(x-(-2))^2$ and $r=\sqrt{16}=4$.

7. What is the completed square form of $y^2+8y$ after adding the needed constant?

   $y^2+8y+16=(y+4)^2$. Add $16$ to complete: $y^2+8y+16$.

8. What is the standard form after completing the square for $x^2+y^2+8x+2y-8=0$?

   $(x+4)^2+(y+1)^2=25$. Complete square: $(x+4)^2-16+(y+1)^2-1-8=0$.

9. What are the center and radius of $x^2+y^2+4x-10y+13=0$?

   Center $(-2,5)$, radius $4$. From $(x+2)^2+(y-5)^2=16$, center $(-2,5)$, $r=4$.

10. What is the standard form of $x^2+y^2+6x-4y+9=0$?

    $(x+3)^2+(y-2)^2=4$. From general form after completing the square.

11. What is the equation of the circle with center $ (0,-3) $ and radius $\sqrt{10}$?

    $x^2+(y+3)^2=10$. Center $ (0,-3) $ gives $x^2+(y+3)^2=(\sqrt{10})^2$.

12. What are the center and radius of $x^2+y^2+12x-16y+36=0$?

    Center $(-6,8)$, radius $8$. From $(x+6)^2+(y-8)^2=64$, $r=8$.

13. What is the standard form of $x^2+y^2-2x+4y+1=0$?

    $(x-1)^2+(y+2)^2=4$. From general form after completing the square.

14. What is the standard form after completing the square for $x^2+y^2-10x-4y+13=0$?

    $(x-5)^2+(y-2)^2=16$. Complete square: $(x-5)^2-25+(y-2)^2-4+13=0$.

15. What is the standard form of $x^2+y^2+10x-2y+1=0$?

    $(x+5)^2+(y-1)^2=25$. From general form after completing the square.

16. What theorem justifies $ $(x-h)^2$+$(y-k)^2$=$r^2$ $ as the circle equation in the coordinate plane?

    The Pythagorean Theorem. Distance formula is based on the Pythagorean Theorem.

17. What are the center and radius of $x^2+y^2-6x+8y-11=0$?

    Center $(3,-4)$, radius $6$. From completed square form $(x-3)^2+(y+4)^2=36$.

18. What are the center and radius of $x^2+y^2-10x-4y+13=0$?

    Center $(5,2)$, radius $4$. From $(x-5)^2+(y-2)^2=16$, $r=\sqrt{16}=4$.

19. What are the center and radius of $x^2+y^2+6x-2y-6=0$?

    Center $(-3,1)$, radius $4$. From $(x+3)^2+(y-1)^2=16$, $r=4$.

20. What is the center and radius of $x^2+y^2+2x-8y+5=0$?

    Center $(-1,4)$, radius $\sqrt{12}$. Complete square: $(x+1)^2-1+(y-4)^2-16+5=0$.

21. What are the center and radius of $x^2+y^2-12x+2y+5=0$?

    Center $(6,-1)$, radius $\sqrt{32}$. From $(x-6)^2+(y+1)^2=32$, $r=\sqrt{32}$.

22. What is the center and radius of $x^2+y^2+6x-4y+9=0$?

    Center $(-3,2)$, radius $2$. Complete square: $(x+3)^2-9+(y-2)^2-4+9=0$.

23. What is the standard form after completing the square for $x^2+y^2-6x+8y-11=0$?

    $(x-3)^2+(y+4)^2=36$. Complete square for $x$ and $y$ terms separately.

24. What is the center and radius of the circle $(x-3)^2+(y+5)^2=49$?

    Center $(3,-5)$, radius $7$. Read center from $(x-h)^2+(y-k)^2=r^2$ and find $r=\sqrt{49}=7$.

25. What is the center and radius of the circle $ $(x-3)^2$ + $(y+5)^2$ = 49 $?

    Center $ (3, -5) $, radius $ 7 $. Read center from $ $(x-h)^2$ + $(y-k)^2$ = $r^2$ $ and find $r = \sqrt{49} = 7$.

26. What is the standard form after completing the square for $x^2+y^2-8x+6y-11=0$?

    $(x-4)^2+(y+3)^2=36$. Complete square: $(x-4)^2-16+(y+3)^2-9-11=0$.

27. What is the center and radius of $x^2+y^2=81$?

    Center $(0,0)$, radius $9$. Circle centered at origin with $r=\sqrt{81}=9$.

28. Identify the geometric figure given by $(x-1)^2+(y+9)^2=0$.

    A single point at $(1,-9)$. Circle with radius $0$ is just the center point.

29. What is the standard form after completing the square for $x^2+y^2+4x-10y+13=0$?

    $(x+2)^2+(y-5)^2=16$. Complete square: $(x+2)^2-4+(y-5)^2-25+13=0$.

30. What is the center and radius of $x^2+y^2=\frac{1}{4}$?

    Center $(0,0)$, radius $\frac{1}{2}$. Origin center with $r=\sqrt{\frac{1}{4}}=\frac{1}{2}$.

31. What is the expanded general form of $(x-h)^2+(y-k)^2=r^2$ called?

    General form $x^2+y^2+Dx+Ey+F=0$. Expanded form with linear terms and constant.

32. What is the standard form after completing the square for $x^2+y^2-2x-12y+20=0$?

    $(x-1)^2+(y-6)^2=17$. Complete square: $(x-1)^2-1+(y-6)^2-36+20=0$.

33. What is the equation of a circle with center $(4,-1)$ and radius $6$?

    $(x-4)^2+(y+1)^2=36$. Substitute $(h,k)=(4,-1)$ and $r^2=36$ into standard form.

34. What is the standard form after completing the square for $x^2+y^2+6x-2y-6=0$?

    $(x+3)^2+(y-1)^2=16$. Complete square: $(x+3)^2-9+(y-1)^2-1-6=0$.

35. What is the standard form after completing the square for $x^2+y^2+2x+2y-7=0$?

    $(x+1)^2+(y+1)^2=9$. Complete square: $(x+1)^2-1+(y+1)^2-1-7=0$.

36. What is the standard form after completing the square for $x^2+y^2+12x-16y+36=0$?

    $(x+6)^2+(y-8)^2=64$. Complete square: $(x+6)^2-36+(y-8)^2-64+36=0$.

37. What number is added to complete the square for $y^2-6y$?

    Add $9$. Take half of coefficient: $(\frac{-6}{2})^2=9$.

38. What number is added to complete the square for $x^2+10x$?

    Add $25$. Take half of coefficient: $(\frac{10}{2})^2 = 25$.

39. What are the center and radius of $x^2+y^2-8x+6y-11=0$?

    Center $(4,-3)$, radius $6$. From $(x-4)^2+(y+3)^2=36$, $r=6$.

40. What is the standard form equation of a circle with center $(h,k)$ and radius $r$?

    $(x-h)^2+(y-k)^2=r^2$. Standard form derived from the distance formula.

41. Identify the center and radius from $(x+7)^2+(y-2)^2=\frac{9}{4}$.

    Center $(-7,2)$, radius $\frac{3}{2}$. From $(x+7)^2+(y-2)^2=\frac{9}{4}$, $r=\frac{3}{2}$.

42. Identify the radius of $(x-1)^2+(y+9)^2=0$.

    Radius $0$. From $(x-1)^2+(y+9)^2=0$, radius is $\sqrt{0}=0$.

43. What is the completed square form of $x^2-14x$ after adding the needed constant?

    $x^2-14x+49=(x-7)^2$. Add $49$ to complete: $x^2-14x+49$.

44. What is the equation of a circle with center $(-3,2)$ and radius $5$?

    $(x+3)^2+(y-2)^2=25$. Use $(h,k)=(-3,2)$ and $r^2=25$ in standard form.

45. What is the distance formula between $(x,y)$ and $(h,k)$ used to derive a circle equation?

    $d=\sqrt{(x-h)^2+(y-k)^2}$. Square root of sum of squared differences in coordinates.

46. What is the center and radius of $x^2+y^2-2x+4y+1=0$?

    Center $(1,-2)$, radius $2$. Complete square: $(x-1)^2-1+(y+2)^2-4+1=0$.

47. What are the center and radius of $x^2+y^2-2x-12y+20=0$?

    Center $(1,6)$, radius $\sqrt{17}$. From standard form, $r=\sqrt{17}$ (cannot simplify).

48. What are the center and radius of $x^2+y^2+2x+2y-7=0$?

    Center $(-1,-1)$, radius $3$. From $(x+1)^2+(y+1)^2=9$, $r=3$.

49. What is the equation of the circle with center $(-5,0)$ and radius $\frac{1}{2}$?

    $(x+5)^2+y^2=\frac{1}{4}$. Center $(-5,0)$ gives $(x+5)^2+y^2=(\frac{1}{2})^2$.

50. What are the center and radius of $x^2+y^2-4x+14y+29=0$?

    Center $(2,-7)$, radius $\sqrt{24}$. From $(x-2)^2+(y+7)^2=24$, $r=\sqrt{24}$.

51. What equation results from setting the distance from $(x,y)$ to $(h,k)$ equal to $r$ and squaring?

    $(x-h)^2+(y-k)^2=r^2$. Setting distance equal to $r$ and squaring both sides.

52. What is the center and radius of $x^2+y^2+10x-2y+1=0$?

    Center $(-5,1)$, radius $5$. Complete square: $(x+5)^2-25+(y-1)^2-1+1=0$.

53. What is the standard form after completing the square for $x^2+y^2-12x+2y+5=0$?

    $(x-6)^2+(y+1)^2=32$. Complete square: $(x-6)^2-36+(y+1)^2-1+5=0$.

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