Derive the Equation of a Circle - Algebra 2

1. What is the general form after expanding $(x-2)^2+(y+3)^2=16$?

   $x^2+y^2-4x+6y-3=0$. Expand and rearrange to general form.

2. What is the radius of a circle with center $(1,-2)$ passing through $(4,2)$?

   $5$. Distance formula: $\sqrt{(4-1)^2+(2-(-2))^2}=5$.

3. What is the center of the circle $x^2+y^2-2x+14y+33=0$?

   $(1,-7)$. From standard form, center is $(1,-7)$.

4. What is the equation of the circle centered at $(-2,1)$ passing through $(1,5)$?

   $(x+2)^2+(y-1)^2=25$. Distance from $(-2,1)$ to $(1,5)$ is $r=5$.

5. What is the equation of the circle with center $(0,7)$ and radius $2$?

   $x^2+(y-7)^2=4$. Substitute center $(h,k)=(0,7)$ and radius $r=2$.

6. What is the equation of the circle centered at $(0,0)$ passing through $(-3,4)$?

   $x^2+y^2=25$. Distance from origin to $(-3,4)$ is $r=5$.

7. What is the center of the circle $(x-3)^2+(y+5)^2=16$?

   $(3,-5)$. From $(x-3)^2+(y+5)^2$, center is $(3,-5)$.

8. What is the completed-square form of $x^2+Dx$?

   $\left(x+\frac{D}{2}\right)^2-\left(\frac{D}{2}\right)^2$. Perfect square trinomial minus the added constant.

9. What is the general form after expanding $(x+5)^2+(y-1)^2=36$?

   $x^2+y^2+10x-2y-10=0$. Expand and rearrange to general form.

10. What is the standard form of $x^2+y^2-2x+14y+33=0$?

    $(x-1)^2+(y+7)^2=17$. Complete square with $D=-2$, $E=14$, $F=33$.

11. What is the completed-square form of $y^2+Ey$?

    $\left(y+\frac{E}{2}\right)^2-\left(\frac{E}{2}\right)^2$. Perfect square trinomial for $y$ terms.

12. What is the equation of the circle with center $(-4,6)$ and radius $3$?

    $(x+4)^2+(y-6)^2=9$. Substitute center $(h,k)=(-4,6)$ and radius $r=3$.

13. What is the general form after expanding $x^2+(y-4)^2=20$?

    $x^2+y^2-8y-4=0$. Expand and rearrange to general form.

14. What is the center and radius of $(x+1)^2+(y-2)^2=0$?

    Center $(-1,2)$; radius $0$. Degenerate circle: single point at $(-1,2)$.

15. What is the completed-square expression equivalent to $y^2+8y$?

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

16. What is the center and radius of $x^2+y^2+2x+6y+1=0$?

    Center $(-1,-3)$; radius $3$. Complete square: $D=2$, $E=6$, $F=1$.

17. What is the radius in terms of diameter endpoints $(x_1,y_1)$ and $(x_2,y_2)$?

    $r=\frac{1}{2}\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$. Radius is half the distance between endpoints.

18. What midpoint formula gives the center of a circle from diameter endpoints $(x_1,y_1)$ and $(x_2,y_2)$?

    $\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$. Midpoint formula gives center from endpoints.

19. What is the completed-square expression equivalent to $x^2-12x$?

    $(x-6)^2-36$. Complete square: $(x-6)^2=x^2-12x+36$.

20. What is the radius of the circle with diameter endpoints $(2,-1)$ and $(8,5)$?

    $3\sqrt{2}$. Half the distance between diameter endpoints.

21. What is the center of $x^2+y^2+Dx+Ey+F=0$ after completing the square?

    $\left(-\frac{D}{2},-\frac{E}{2}\right)$. Center coordinates are negatives of linear coefficients.

22. 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 shows center $(h,k)$ and radius $r$ explicitly.

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

    $(x-4)^2+(y+1)^2=25$. Complete square with $D=-8$, $E=2$, $F=-8$.

24. What is the equation of the circle with diameter endpoints $(-2,4)$ and $(4,-2)$?

    $(x-1)^2+(y-1)^2=18$. Midpoint $(1,1)$, radius half diagonal length.

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

    Center $(-2,5)$; radius $4$. Complete square: $D=4$, $E=-10$, $F=13$.

26. What is the Pythagorean distance formula between $(x,y)$ and $(h,k)$?

    $d=\sqrt{(x-h)^2+(y-k)^2}$. Distance formula derived from Pythagorean theorem.

27. Identify the value added to complete the square for $x^2-14x$.

    $49$. Take half of coefficient: $(-14/2)^2=49$.

28. What is the standard form of $x^2+y^2+10x+12y+9=0$?

    $(x+5)^2+(y+6)^2=52$. Complete square with $D=10$, $E=12$, $F=9$.

29. What must be true about the coefficients of $x^2$ and $y^2$ for a circle in general form?

    They are equal and nonzero. Both coefficients must be 1 for a true circle.

30. What is the equation of the circle with diameter endpoints $(0,0)$ and $(6,8)$?

    $(x-3)^2+(y-4)^2=25$. Midpoint $(3,4)$, radius half diagonal length.

31. What is the standard form of $x^2+y^2+6x-4y-12=0$?

    $(x+3)^2+(y-2)^2=25$. Complete square with $D=6$, $E=-4$, $F=-12$.

32. What is the general form of a circle equation after expanding standard form?

    $x^2+y^2+Dx+Ey+F=0$. Result of expanding standard form and collecting terms.

33. What is the radius of the circle $x^2+y^2+10x+12y+9=0$?

    $\sqrt{52}$. From standard form, $r^2=52$, so $r=\sqrt{52}$.

34. What is the equation of a circle with center $(h,k)$ passing through $(x_1,y_1)$?

    $(x-h)^2+(y-k)^2=(x_1-h)^2+(y_1-k)^2$. Distance from center to point equals the radius.

35. What is the equation of a circle centered at the origin with radius $r$?

    $x^2+y^2=r^2$. When center is at origin, $h=0$ and $k=0$.

36. What is the center and radius of the circle $(x-h)^2+(y-k)^2=r^2$?

    Center $(h,k)$; radius $r$. Read center and radius directly from standard form.

37. What is the radius of the circle $x^2+y^2=49$?

    $7$. Since $r^2=49$, take square root to get $r=7$.

38. What does it mean if a circle equation gives $r^2=0$ after completing the square?

    The circle is a single point (a degenerate circle). Circle degenerates to single point when $r^2=0$.

39. What is the center and radius of $x^2+y^2-12x-2y+37=0$?

    Center $(6,1)$; radius $0$. Complete square: $r^2=36+1-37=0$.

40. What is the equation of the circle with center $(2,-1)$ and radius $5$?

    $(x-2)^2+(y+1)^2=25$. Substitute center $(h,k)=(2,-1)$ and radius $r=5$.

41. What is $r^2$ for $x^2+y^2+Dx+Ey+F=0$ in terms of $D,E,F$?

    $\left(\frac{D}{2}\right)^2+\left(\frac{E}{2}\right)^2-F$. Sum of completing square constants minus $F$.

42. What is $r^2$ for a circle centered at $(h,k)$ passing through $(x_1,y_1)$?

    $r^2=(x_1-h)^2+(y_1-k)^2$. Radius squared equals distance squared to point.

43. Identify the value added to complete the square for $y^2+9y$.

    $\frac{81}{4}$. Take half of coefficient: $(9/2)^2=81/4$.

44. What is the equation of the circle in standard form: $x^2+y^2=9$?

    $(x-0)^2+(y-0)^2=9$. Already in standard form with center $(0,0)$.

45. What is the equation of the circle with center $(-3,0)$ and radius $10$?

    $(x+3)^2+y^2=100$. Substitute center $(h,k)=(-3,0)$ and radius $r=10$.

46. What is the first step to complete the square in $x^2+Dx$?

    Add and subtract $\left(\frac{D}{2}\right)^2$. Add $\left(\frac{D}{2}\right)^2$ to create perfect square trinomial.

47. What is the radius of the circle $(x-3)^2+(y+5)^2=16$?

    $4$. Since $r^2=16$, take square root to get $r=4$.

48. What is the center of the circle with diameter endpoints $(2,-1)$ and $(8,5)$?

    $(5,2)$. Midpoint formula for diameter endpoints.

49. What does it mean if completing the square gives $r^2<0$?

    No real circle exists (no real points satisfy it). Negative $r^2$ means no real solutions exist.

50. What equation results from setting distance from $(x,y)$ to $(h,k)$ equal to $r$?

    $(x-h)^2+(y-k)^2=r^2$. Circle equation comes from distance formula equal to $r$.

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

    Center $(3,-4)$; radius $6$. Complete square: $D=-6$, $E=8$, $F=-11$.

52. What is the equation of the circle centered at $(3,4)$ passing through $(6,8)$?

    $(x-3)^2+(y-4)^2=25$. Distance from $(3,4)$ to $(6,8)$ is $r=5$.

53. What is the center and radius of $(x-7)^2+(y+3)^2=81$?

    Center $(7,-3)$; radius $9$. Read center and radius from standard form.

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