Rotated Conic Equation Calculator

FAQs

1. How do you solve a rotated conic? Solving a rotated conic involves identifying the type of conic section (ellipse, hyperbola, or parabola) and finding its equation in standard form. Then, you can use various techniques such as completing the square or algebraic manipulation to solve for specific parameters.

2. What is the equation of a rotated parabola? The equation of a rotated parabola depends on the specific rotation and orientation. It would involve trigonometric functions and translations. The general form is complex and depends on the angle of rotation.

3. How do you rotate a hyperbola equation? To rotate a hyperbola equation, you can use a coordinate transformation. You’ll need to substitute new variables that are linear combinations of the original variables and then express the hyperbola’s equation in terms of these new variables.

4. What is the eccentricity of a rotated conic? The eccentricity of a rotated conic depends on the type of conic section and its orientation. Ellipses have eccentricities between 0 and 1, hyperbolas have eccentricities greater than 1, and parabolas have eccentricity equal to 1. Rotating the conic affects the shape but not the range of eccentricity.

5. How do you find the equation of a rotated ellipse? To find the equation of a rotated ellipse, you’ll need to use a coordinate transformation to align the axes with the major and minor axes of the ellipse. This transformation will involve trigonometric functions and translations.

6. What is a rotated parabola called? A rotated parabola is not given a specific name; it is still referred to as a parabola but in a rotated or tilted orientation.

7. What is the equation of a parabola rotated 90 degrees? The equation of a parabola rotated 90 degrees would depend on the specific orientation and axis of rotation. It would be a complex equation involving trigonometric functions.

8. Can a parabola be rotated? Yes, a parabola can be rotated by applying a coordinate transformation to change its orientation relative to the standard vertical or horizontal orientation.

9. How do you graph a rotated parabola? To graph a rotated parabola, you’ll need to know the equation of the parabola after rotation. Plot key points, such as the vertex, focus, and directrix, and then sketch the parabola accordingly.

10. What is rotation in hyperbolic geometry? In hyperbolic geometry, rotation is a transformation that preserves angles and distances in the hyperbolic plane. It differs from Euclidean geometry because the angles of rotation are not the same as in Euclidean geometry.

11. What is the rotation of a rectangular hyperbola? The rotation of a rectangular hyperbola can change its orientation, making it oblique relative to the standard horizontal and vertical axes.

12. What is the formula for solving a hyperbola? The formula for solving a hyperbola depends on the specific problem, such as finding its asymptotes, foci, vertices, or eccentricity. Each problem involves different equations and techniques.

13. What is the eccentricity of a rotated ellipse? The eccentricity of a rotated ellipse remains the same as the eccentricity of the original ellipse. It is a measure of how stretched or elongated the ellipse is.

14. What is the rotation of axes and second-degree equation? The rotation of axes is a technique used to simplify second-degree equations (conic sections) by aligning the coordinate axes with the major and minor axes of the conic section. This simplification makes it easier to identify the conic’s standard form.

15. What is a conic if eccentricity is 1? If the eccentricity of a conic section is exactly 1, it represents a parabola.

16. What is the general equation for rotation in equation form? The general equation for rotating a point (x, y) by an angle θ counterclockwise around the origin is: x’ = xcos(θ) – ysin(θ) y’ = xsin(θ) + ycos(θ)

17. How do you find the foci of a rotated ellipse? To find the foci of a rotated ellipse, first determine the equation of the ellipse in standard form. Then, use the standard formulas for the foci of an ellipse.

18. What is the general equation of a conic section? The general equation of a conic section in standard form depends on the type of conic:

• For an ellipse: (x^2/a^2) + (y^2/b^2) = 1
• For a hyperbola: (x^2/a^2) – (y^2/b^2) = 1 (hyperbola opens horizontally)
• For a hyperbola: (y^2/a^2) – (x^2/b^2) = 1 (hyperbola opens vertically)
• For a parabola: (x^2/a^2) = 4py (parabola opens upward or downward)
• For a parabola: (y^2/a^2) = 4px (parabola opens rightward or leftward)

19. What is the formula for the elliptic paraboloid? The formula for the elliptic paraboloid is: z = (x^2/a^2) + (y^2/b^2)

20. What happens when a figure is rotated 90 degrees? When a figure is rotated 90 degrees counterclockwise, its orientation changes, and its points move 90 degrees around a specified center of rotation.

21. What is the rule for rotating 90 degrees on a graph? To rotate a figure 90 degrees counterclockwise on a graph, you can switch the x and y coordinates and negate the new x coordinate.

22. How do you rotate a point on a graph 90 degrees clockwise? To rotate a point 90 degrees clockwise on a graph, you can switch the x and y coordinates, negate the new y coordinate, and keep the same sign for the new x coordinate.

23. Can you rotate a quadratic equation? Yes, you can rotate a quadratic equation by using a coordinate transformation, but the resulting equation will be more complex and involve trigonometric functions.

24. Are parabolas always curved? Yes, parabolas are always curved. They are a type of conic section that can open upward or downward, leftward or rightward, but they are always curved.

25. What is the formula for graph rotation? The formula for graph rotation depends on the type of transformation, but for a basic 2D rotation, you can use: x’ = xcos(θ) – ysin(θ) y’ = xsin(θ) + ycos(θ)

26. What are the rules for parabolas? Parabolas have several key characteristics and rules, including their vertex, focus, directrix, axis of symmetry, and the nature of their curvature.

27. What is the 1 3 5 rule for graphing parabolas? The “1-3-5 rule” is not a standard rule for graphing parabolas. It might be a reference to the fact that a parabola has a focus located at a point one unit above the vertex, and the directrix is located three units below the vertex.

28. What are the 4 types of rotation? The four types of rotation are counterclockwise (positive), clockwise (negative), 180-degree (half-turn), and no rotation (zero degrees).

29. What are the 3 types of rotation? The three types of rotation are counterclockwise (positive), clockwise (negative), and 180-degree (half-turn).

30. What is skew rotation? Skew rotation is a rotation that occurs in a non-planar fashion, where the rotation axis is neither parallel nor perpendicular to the plane of rotation. It results in a twisting or skewing of the object.

31. What is the difference between hyperbola and parabola? The main difference between a hyperbola and a parabola is their shape and the number of foci. A hyperbola has two foci and consists of two separate branches, while a parabola has one focus and is a single curve.

32. What is a rectangular hyperbola in a conic section? A rectangular hyperbola is a type of hyperbola where the asymptotes are perpendicular to each other, forming a rectangle in the hyperbola’s center.

33. What shape does a rotating rectangle make? When a rectangle is rotated, it can form various shapes, depending on the angle of rotation. It can create parallelograms, rhombi, or other irregular shapes.

34. What is the equation of a hyperbola in a conic section? The equation of a hyperbola in a conic section is typically one of the following:

• Horizontal hyperbola: (x^2/a^2) – (y^2/b^2) = 1
• Vertical hyperbola: (y^2/a^2) – (x^2/b^2) = 1

35. What is the general equation of a hyperbola example? A general equation of a hyperbola could be (x^2/a^2) – (y^2/b^2) = 1, where ‘a’ and ‘b’ are constants defining the shape and size of the hyperbola.

36. What is the rule for hyperbola? The rule for a hyperbola involves the difference between distances from any point on the hyperbola to its two foci being constant and equal to 2a, where ‘a’ is the distance from the center to a vertex.

37. Is the major axis A or B? The major axis of an ellipse or hyperbola is typically represented by ‘2a,’ where ‘a’ is the semi-major axis. In the equations, ‘a’ is associated with the major axis, and ‘b’ is associated with the minor axis.

38. What is the formula for the eccentricity of a conic section? The formula for the eccentricity (e) of a conic section (ellipse or hyperbola) is given by: e = √(1 + (b^2/a^2))

39. What is the relationship between major and minor axis? The major axis is longer than the minor axis in an ellipse or hyperbola, and their lengths are related by ‘2a’ for the major axis and ‘2b’ for the minor axis.

40. What is the rule for rotations? The rule for rotations involves using trigonometric functions (sine and cosine) to transform the coordinates of points around a fixed center of rotation.

41. What is the two reflection theorem for rotation? The two reflection theorem states that any rotation can be achieved by reflecting the object twice, once in each of two intersecting lines.

42. What is the formula for rotated about the Y axis? Rotating a point about the Y-axis involves changing the x and z coordinates while leaving the y coordinate unchanged. The general formula is: x’ = xcos(θ) + zsin(θ) z’ = -xsin(θ) + zcos(θ)

43. What happens when eccentricity is 0? When the eccentricity of a conic section is 0, it represents a circle. In a circle, all points are equidistant from the center.

44. Why is circle eccentricity zero? The eccentricity of a circle is zero because it has a perfectly circular shape, with all points equidistant from the center. This results in the numerator of the eccentricity formula being zero.

45. Is the eccentricity of a parabola always 1? Yes, the eccentricity of a parabola is always equal to 1. This is a defining characteristic of parabolas.

46. What are the 5 rotation formulas? The 5 rotation formulas likely refer to the following:

1. Rotation of a point in 2D: x’ = xcos(θ) – ysin(θ) and y’ = xsin(θ) + ycos(θ)
2. Rotation of a point in 3D about the X-axis.
3. Rotation of a point in 3D about the Y-axis.
4. Rotation of a point in 3D about the Z-axis.
5. Rotation of a point in 3D about an arbitrary axis using the Rodrigues’ rotation formula.

47. What is the tilt angle of an ellipse? The tilt angle of an ellipse is the angle by which it is rotated or tilted from its standard orientation, where the major and minor axes are aligned with the coordinate axes.

48. Can an ellipse be rotated? Yes, an ellipse can be rotated by changing its orientation relative to the standard horizontal and vertical axes.

49. What is the order of rotation of an ellipse? The order of rotation of an ellipse is determined by the angle by which it is rotated counterclockwise. The angle of rotation is typically measured in degrees or radians.

50. What is the rotational symmetry of ellipse? An ellipse has rotational symmetry if it can be rotated by an angle less than 360 degrees and still appear the same. Ellipses have rotational symmetry if they are perfectly symmetric, meaning their axes are equal in length.

51. What is the equation for a rotated paraboloid? The equation for a rotated paraboloid in 3D space would depend on the specific rotation and orientation of the paraboloid. It would involve trigonometric functions and translations.

52. What is the general equation of elliptic curve? The general equation of an elliptic curve is given by: y^2 = x^3 + ax + b Where ‘a’ and ‘b’ are constants that determine the specific characteristics of the elliptic curve.