Derive Equations of Ellipses and Parabolas - Geometry

1. What are the vertices of a vertical hyperbola centered at $(h,k)$?

   $(h,\,k\pm a)$. Vertices are the endpoints of the transverse axis, $a$ units from center.

2. Find the center if the foci are $(-2,5)$ and $(8,5)$.

   $(3,\,5)$. The center is the midpoint: $\left(\frac{-2+8}{2}, \frac{5+5}{2}\right) = (3,5)$.

3. What are the foci of a horizontal ellipse centered at $(h,k)$?

   $(h\pm c,\,k)$. Horizontal ellipses have foci along the $x$-axis, displaced $c$ units from center.

4. What are the vertices of a horizontal hyperbola centered at $(h,k)$?

   $(h\pm a,\,k)$. Vertices are the endpoints of the transverse axis, $a$ units from center.

5. Find $e$ for a hyperbola with $a=5$ and $c=13$.

   $e=\frac{13}{5}$. Eccentricity: $e = \frac{c}{a} = \frac{13}{5}$.

6. What is the equation of the hyperbola with foci $(0, \pm 6)$ and constant difference $4$?

   $\frac{y^2}{4}-\frac{x^2}{32}=1$. Vertical hyperbola: $a = 2$, $c = 6$, so $b^2 = 36 - 4 = 32$.

7. What is the locus definition of an ellipse using foci $F_1,F_2$ and constant $2a$?

   $d(P,F_1)+d(P,F_2)=2a$. The sum of distances from any point on an ellipse to both foci equals $2a$.

8. What is the equation of the hyperbola with foci $(-2,1)$ and $(6,1)$ and constant difference $6$?

   $\frac{(x-2)^2}{9}-\frac{(y-1)^2}{7}=1$. Center $(2,1)$, $c = 4$, $a = 3$, horizontal, so $b^2 = 16 - 9 = 7$.

9. What is the standard form of a horizontal ellipse centered at $(h,k)$ with $a>b$?

   $\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1$. For horizontal ellipses, $a$ (semi-major axis) goes under the $x$-term when $a>b$.

10. For a hyperbola centered at $ (h,k) $, how do you find the center from foci $ ($x_1$,$y_1$) $ and $ ($x_2$,$y_2$) $?

    $\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)$. The center is the midpoint of the two foci.

11. What is the equation of the hyperbola with foci $(0,\pm 6)$ and constant difference $4$?

    $\frac{y^2}{4}-\frac{x^2}{32}=1$. Vertical hyperbola: $a = 2$, $c = 6$, so $b^2 = 36 - 4 = 32$.

12. What are the foci of a vertical hyperbola centered at $(h,k)$?

    $(h,\,k\pm c)$. Vertical hyperbolas have foci along the $y$-axis, displaced $c$ units from center.

13. What are the vertices of a vertical ellipse centered at $(h,k)$?

    $(h,\,k\pm a)$. Vertices are the endpoints of the major axis, $a$ units from center.

14. What is the equation of the ellipse with foci $(-1,2)$ and $(7,2)$ and constant sum $14$?

    $\frac{(x-3)^2}{49}+\frac{(y-2)^2}{33}=1$. Center $(3,2)$, $c = 4$, $a = 7$, horizontal, so $b^2 = 49 - 16 = 33$.

15. For an ellipse centered at $(h,k)$, how do you find the center from foci $(x_1,y_1)$ and $(x_2,y_2)$?

    $\left(\frac{x_1+x_2}{2},\,\frac{y_1+y_2}{2}\right)$. The center is the midpoint of the two foci.

16. What is the standard form of a horizontal hyperbola centered at $ (h,k) $?

    $\frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}=1$. Horizontal hyperbolas have the positive term with $x$ and negative term with $y$.

17. What is the standard form of a vertical ellipse centered at $(h,k)$ with $a>b$?

    $\frac{(x-h)^2}{b^2}+\frac{(y-k)^2}{a^2}=1$. For vertical ellipses, $a$ (semi-major axis) goes under the $y$-term when $a>b$.

18. Which equation matches a horizontal hyperbola with center $(h,k)$, $a=3$, and $b=4$?

    $\frac{(x-h)^2}{9}-\frac{(y-k)^2}{16}=1$. Horizontal hyperbola with $a^2 = 9$ and $b^2 = 16$.

19. What is the distance formula between $(x,y)$ and $(h,k)$ in the plane?

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

20. Find $b^2$ for a hyperbola if $a=9$ and $c=15$.

    $b^2=144$. Using $c^2 = a^2 + b^2$: $b^2 = 225 - 81 = 144$.

21. Find $e$ for an ellipse with $a=10$ and $c=6$.

    $e=\frac{3}{5}$. Eccentricity: $e = \frac{c}{a} = \frac{6}{10} = \frac{3}{5}$.

22. What is the equation of the hyperbola with center $(0,0)$, foci $(0,\pm 13)$, and constant difference $10$?

    $\frac{y^2}{25}-\frac{x^2}{144}=1$. Vertical hyperbola: $a = 5$, $c = 13$, so $b^2 = 169 - 25 = 144$.

23. Identify the error: Using $c^2=a^2+b^2$ for an ellipse. What is the correct relation?

    Correct: $c^2=a^2-b^2$. Ellipses use subtraction: $c^2 = a^2 - b^2$ since $c < a$.

24. Find the vertices of the ellipse with foci $(\pm 3,0)$ and constant sum $10$.

    $(\pm 5,\,0)$. From constant sum 10: $a = 5$, so vertices are at $(\pm 5, 0)$.

25. What are the foci of a horizontal hyperbola centered at $(h,k)$?

    $(h\pm c,\,k)$. Horizontal hyperbolas have foci along the $x$-axis, displaced $c$ units from center.

26. Identify the error: Using $c^2=a^2-b^2$ for a hyperbola. What is the correct relation?

    Correct: $c^2=a^2+b^2$. Hyperbolas use addition: $c^2 = a^2 + b^2$ since $c > a$.

27. Identify the value of $a$ for a hyperbola if the constant difference of distances is $10$.

    $a=5$. The constant difference $2a = 10$, so $a = 5$.

28. Find $c$ for an ellipse if $a=13$ and $b=5$.

    $c=12$. Using $c^2 = a^2 - b^2$: $c^2 = 169 - 25 = 144$, so $c = 12$.

29. For a hyperbola centered at $(h,k)$, how do you find the center from foci $(x_1,y_1)$ and $(x_2,y_2)$?

    $\left(\frac{x_1+x_2}{2},\,\frac{y_1+y_2}{2}\right)$. The center is the midpoint of the two foci.

30. What is the standard form of a vertical hyperbola centered at $(h,k)$?

    $\frac{(y-k)^2}{a^2}-\frac{(x-h)^2}{b^2}=1$. Vertical hyperbolas have the positive term with $y$ and negative term with $x$.

31. What is the eccentricity formula for an ellipse or hyperbola in terms of $a$ and $c$?

    $e=\frac{c}{a}$. Eccentricity measures how elongated the conic is relative to a circle.

32. What is the asymptote form for a vertical hyperbola $\frac{(y-k)^2}{a^2}-\frac{(x-h)^2}{b^2}=1$?

    $y-k=\pm \frac{a}{b}(x-h)$. For vertical hyperbolas, slopes are $\pm \frac{a}{b}$.

33. What is the equation of the ellipse with center $(0,0)$, foci $(\pm 3,0)$, and constant sum $10$?

    $\frac{x^2}{25}+\frac{y^2}{16}=1$. Horizontal ellipse: $a = 5$, $c = 3$, so $b^2 = 25 - 9 = 16$.

34. Which equation matches a vertical ellipse with center $(h,k)$, $a=7$, and $b=2$?

    $\frac{(x-h)^2}{4}+\frac{(y-k)^2}{49}=1$. Vertical ellipse with $a^2 = 49$ and $b^2 = 4$.

35. What is the equation of the hyperbola with center $(0,0)$, foci $(\pm 5,0)$, and constant difference $6$?

    $\frac{x^2}{9}-\frac{y^2}{16}=1$. Horizontal hyperbola: $a = 3$, $c = 5$, so $b^2 = 25 - 9 = 16$.

36. Find $b^2$ for an ellipse if $a=10$ and $c=6$.

    $b^2=64$. Using $c^2 = a^2 - b^2$: $b^2 = 100 - 36 = 64$.

37. For a hyperbola, what relationship connects $a$, $b$, and focal distance $c$?

    $c^2=a^2+b^2$. For hyperbolas, $c>a$ so $c^2 = a^2 + b^2$ where $c$ is the focal distance.

38. Identify the transverse axis orientation if the foci are $(h\pm c,k)$.

    Horizontal. Foci with same $y$-coordinate indicate horizontal transverse axis.

39. What is the equation of the ellipse with foci $(\pm 2,0)$ and constant sum $12$?

    $\frac{x^2}{36}+\frac{y^2}{32}=1$. Horizontal ellipse: $a = 6$, $c = 2$, so $b^2 = 36 - 4 = 32$.

40. What is the equation of the hyperbola with foci $ (5,-4) $ and $ (5,8) $ and constant difference $ 8 $?

    $\frac{(y-2)^2}{16}-\frac{(x-5)^2}{20}=1$. Center $ (5,2) $, $ c = 6 $, $ a = 4 $, vertical, so $ $b^2$ = 36 - 16 = 20 $.

41. Find the center if the foci are $(4,-7)$ and $(4,9)$.

    $(4,\,1)$. The center is the midpoint: $\left(\frac{4+4}{2}, \frac{-7+9}{2}\right) = (4,1)$.

42. What is the equation of the ellipse with foci $(4,-3)$ and $(4,9)$ and constant sum $20$?

    $\frac{(x-4)^2}{64}+\frac{(y-3)^2}{100}=1$. Center $(4,3)$, $c = 6$, $a = 10$, vertical, so $b^2 = 100 - 36 = 64$.

43. For an ellipse, what inequality must hold between $2a$ and the distance between the foci $2c$?

    $2a>2c$. For an ellipse to exist, the sum must exceed the distance between foci.

44. For an ellipse, what relationship connects $a$, $b$, and focal distance $c$?

    $c^2=a^2-b^2$. For ellipses, $c<a$ so $c^2 = a^2 - b^2$ where $c$ is the focal distance.

45. What is the equation of the ellipse with center $(0,0)$, foci $(0,\pm 4)$, and constant sum $14$?

    $\frac{x^2}{33}+\frac{y^2}{49}=1$. Vertical ellipse: $a = 7$, $c = 4$, so $b^2 = 49 - 16 = 33$.

46. For a hyperbola, what inequality must hold between $2a$ and the distance between the foci $2c$?

    $2a<2c$. For a hyperbola to exist, the difference must be less than the distance between foci.

47. Find $c$ if the foci of a horizontal conic are $(h-6,k)$ and $(h+6,k)$.

    $c=6$. The distance from center to each focus is $c = 6$.

48. Identify the value of $a$ for an ellipse if the constant sum of distances is $14$.

    $a=7$. The constant sum $2a = 14$, so $a = 7$.

49. What is the locus definition of a hyperbola using foci $F_1,F_2$ and constant $2a$?

    $|d(P,F_1)-d(P,F_2)|=2a$. The absolute difference of distances from any point on a hyperbola to both foci equals $2a$.

50. What condition on eccentricity $e$ indicates a hyperbola?

    $e>1$. Hyperbolas have eccentricity greater than 1.

51. What condition on eccentricity $e$ indicates an ellipse (not a circle)?

    $0<e<1$. Ellipses have eccentricity between 0 and 1 (circle has $e = 0$).

52. Identify the transverse axis orientation if the foci are $(h,k\pm c)$.

    Vertical. Foci with same $x$-coordinate indicate vertical transverse axis.

53. What is the asymptote form for a horizontal hyperbola $\frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}=1$?

    $y-k=\pm \frac{b}{a}(x-h)$. For horizontal hyperbolas, slopes are $\pm \frac{b}{a}$.

54. What is the equation of the hyperbola with foci $(5,-4)$ and $(5,8)$ and constant difference $8$?

    $\frac{(y-2)^2}{16}-\frac{(x-5)^2}{20}=1$. Center $(5,2)$, $c = 6$, $a = 4$, vertical, so $b^2 = 36 - 16 = 20$.

55. Find $c$ for a hyperbola if $a=8$ and $b=6$.

    $c=10$. Using $c^2 = a^2 + b^2$: $c^2 = 64 + 36 = 100$, so $c = 10$.

56. What are the vertices of a horizontal ellipse centered at $(h,k)$?

    $(h\pm a,\,k)$. Vertices are the endpoints of the major axis, $a$ units from center.

57. What are the foci of a vertical ellipse centered at $(h,k)$?

    $(h,\,k\pm c)$. Vertical ellipses have foci along the $y$-axis, displaced $c$ units from center.

58. Find the vertices of the hyperbola with foci $(0,\pm 13)$ and constant difference $10$.

    $(0,\,\pm 5)$. From constant difference 10: $a = 5$, so vertices are at $(0, \pm 5)$.

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