Podcast
Questions and Answers
What occurs to the coordinates of a point when a new set of axes is taken?
What occurs to the coordinates of a point when a new set of axes is taken?
What is the purpose of translating the origin of coordinates?
What is the purpose of translating the origin of coordinates?
How do the equations of transformation relate to the coordinates of a point P?
How do the equations of transformation relate to the coordinates of a point P?
What mathematical relationships are used for rotation of axes?
What mathematical relationships are used for rotation of axes?
Signup and view all the answers
What is a key property of coordinates when transforming to a new set of axes?
What is a key property of coordinates when transforming to a new set of axes?
Signup and view all the answers
In the transformation of coordinates, what does the variable θ represent?
In the transformation of coordinates, what does the variable θ represent?
Signup and view all the answers
Which equation represents the transformation from old to new axes during translation?
Which equation represents the transformation from old to new axes during translation?
Signup and view all the answers
What geometric shape does the transformation of coordinates imply?
What geometric shape does the transformation of coordinates imply?
Signup and view all the answers
What is the correct expression for the transformed equation after substitution?
What is the correct expression for the transformed equation after substitution?
Signup and view all the answers
Which is the correct relationship derived from the transformed coefficients?
Which is the correct relationship derived from the transformed coefficients?
Signup and view all the answers
What condition must be satisfied for the term xy to be absent in the transformed equation?
What condition must be satisfied for the term xy to be absent in the transformed equation?
Signup and view all the answers
What does the expression Δ denote in terms of coefficients?
What does the expression Δ denote in terms of coefficients?
Signup and view all the answers
What transformation is applied to the variables x and y?
What transformation is applied to the variables x and y?
Signup and view all the answers
What is the formula used to find the angle θ for the rotation of axes?
What is the formula used to find the angle θ for the rotation of axes?
Signup and view all the answers
What invariant remains unchanged under the transformation of the equation?
What invariant remains unchanged under the transformation of the equation?
Signup and view all the answers
During the transformation, how is the coefficient g' expressed?
During the transformation, how is the coefficient g' expressed?
Signup and view all the answers
When the original equation 5x² + 4xy + 5y² - 10 = 0 is transformed, which term is removed?
When the original equation 5x² + 4xy + 5y² - 10 = 0 is transformed, which term is removed?
Signup and view all the answers
What is the significance of the equation a'b' - h'² = ab - h²?
What is the significance of the equation a'b' - h'² = ab - h²?
Signup and view all the answers
How does the expression for a' involve the trigonometric function?
How does the expression for a' involve the trigonometric function?
Signup and view all the answers
What values are assigned to a, b, and h in the problem to verify the transformation?
What values are assigned to a, b, and h in the problem to verify the transformation?
Signup and view all the answers
Which equation derives from substituting the rotated coordinates in the original equation?
Which equation derives from substituting the rotated coordinates in the original equation?
Signup and view all the answers
What is the result of comparing the coefficients in the transformed equation?
What is the result of comparing the coefficients in the transformed equation?
Signup and view all the answers
What characteristic does the quantity A represent in the context of transformations?
What characteristic does the quantity A represent in the context of transformations?
Signup and view all the answers
What substitution is used for x in the transformation process?
What substitution is used for x in the transformation process?
Signup and view all the answers
What is the expression for $x'$ in terms of $x$ and $y$ during the transformations?
What is the expression for $x'$ in terms of $x$ and $y$ during the transformations?
Signup and view all the answers
In the transformation to axes inclined at 45°, what are the values of $cos θ$ and $sin θ$?
In the transformation to axes inclined at 45°, what are the values of $cos θ$ and $sin θ$?
Signup and view all the answers
What is the outcome of substituting $x' = x + α$ and $y' = y + β$ into the general equation of the second degree?
What is the outcome of substituting $x' = x + α$ and $y' = y + β$ into the general equation of the second degree?
Signup and view all the answers
How is $y'$ expressed in terms of $x$ and $y$ during the transformations?
How is $y'$ expressed in terms of $x$ and $y$ during the transformations?
Signup and view all the answers
What remains unchanged in the transformed equation of a curve according to the transformations described?
What remains unchanged in the transformed equation of a curve according to the transformations described?
Signup and view all the answers
When transitioning to axes inclined at an angle, what general structure does the transformation follow?
When transitioning to axes inclined at an angle, what general structure does the transformation follow?
Signup and view all the answers
What is the result when the origin is transferred to point $(α, β)$ and a new angle $θ$ is applied?
What is the result when the origin is transferred to point $(α, β)$ and a new angle $θ$ is applied?
Signup and view all the answers
What can be concluded about the constant term after substituting $α$ and $β$ into the general equation?
What can be concluded about the constant term after substituting $α$ and $β$ into the general equation?
Signup and view all the answers
What is the transformed equation of 9x² + 24xy + 2y² - 6x + 20y + 41 = 0 through the point (-1, 1)?
What is the transformed equation of 9x² + 24xy + 2y² - 6x + 20y + 41 = 0 through the point (-1, 1)?
Signup and view all the answers
Which condition must be satisfied to remove the terms in x and y from the transformed equation?
Which condition must be satisfied to remove the terms in x and y from the transformed equation?
Signup and view all the answers
What is the value of θ when the term in xy is removed by rotating the axes?
What is the value of θ when the term in xy is removed by rotating the axes?
Signup and view all the answers
What is the correct substitution for x when performing the transformation for rotating axes?
What is the correct substitution for x when performing the transformation for rotating axes?
Signup and view all the answers
Which transformation is needed to remove the term of y in the equation 3x - 25y + 41 = 0?
Which transformation is needed to remove the term of y in the equation 3x - 25y + 41 = 0?
Signup and view all the answers
What manual calculation is required for the transformation of y(y - 2a) = ax?
What manual calculation is required for the transformation of y(y - 2a) = ax?
Signup and view all the answers
What is the resulting form of the equation 3x² + 5y² - 3 = 0 after rotating through 45°?
What is the resulting form of the equation 3x² + 5y² - 3 = 0 after rotating through 45°?
Signup and view all the answers
What determines the angle of inclination when transforming axes through the point (a, b)?
What determines the angle of inclination when transforming axes through the point (a, b)?
Signup and view all the answers
Study Notes
Transformation of Coordinates
- The coordinates of a point depend upon the origin and the axes of coordinates chosen.
- The equation of a curve will change if the axes of coordinates are changed.
- The process of changing from one set of axes to another is called Transformation of Coordinates.
Translation of Coordinates
- Translation shifts the origin without changing the direction of the axes.
- If the original axes are OX and OY, and the new axes are O'X' and O'Y' through the new origin O', then:
- The coordinates of O' with respect to the original axes are (α, β).
- The coordinates of any point P are (x, y) with respect to the old axes and (x', y') with respect to the new axes.
- The transformation equations are:
- Old to new: x = x' + α, y = y' + β
- New to old: x' = x - α, y' = y - β
Rotation of Coordinates
- Rotation rotates the axes around the origin.
- If the original axes are OX and OY, and the new axes are OX' and OY' rotated by angle θ, then:
- The coordinates of any point P are (x, y) with respect to the old axes and (x', y') with respect to the new axes.
- The transformation equations are:
- Old to new: x = x' * cos θ - y' * sin θ, y = x' * sin θ + y' * cos θ
- New to old: x' = x * cos θ + y * sin θ, y' = -x * sin θ + y * cos θ
- The transformation equations can be conveniently remembered using a matrix:
| x | y | |
|---|---|---|
| x' | cos θ | -sin θ |
| y' | sin θ | cos θ |
Combined Transformation of Coordinates
- If the origin is translated to (α, β) and the new axes are rotated by angle θ, the combined transformation equations are:
- x = α + x' * cos θ - y' * sin θ
- y = β + x' * sin θ + y' * cos θ
Effect of Transformation on the General Equation of the Second Degree
- The general equation of the second degree is ax² + 2hxy + by² + 2gx + 2fy + c = 0.
-
Translation:
- The terms of the highest degree (ax², 2hxy, by²) remain unchanged.
- The constant term is the result of substituting α for x and β for y in the original expression: F(α, β).
- The transformed equation can be written as: ax² + 2hxy + by² + x * δF/δa + y * δF/δβ + F(α, β) = 0
-
Rotation:
- The terms of the highest degree change.
- The coefficients of the transformed equation are:
- a' = (a + b)/2 + (a - b)/2 * cos 2θ + h * sin 2θ
- b' = (a + b)/2 + (a - b)/2 * cos 2θ - h * sin 2θ
- h' = h * cos 2θ - (a - b)/2 * sin 2θ
- g' = g * cos θ + f * sin θ
- f' = -g * sin θ + f * cos θ
- c' = c
-
Invariants:
- Quantities that remain unchanged under all possible transformations are called invariants.
- For the general equation of the second degree, the following are invariants:
- a' + b' = a + b
- a'b' - h'² = ab - h²
- Δ' = Δ (abc + 2fgh - af² - bg² - ch²)
Removing the xy Term
- To remove the xy term in the general equation, rotate the axes through an angle θ = (1/2) * tan⁻¹(2h/(a - b)).
- In this case, h' = 0, and the transformed equation will have no xy term.
Removing the x and y Terms
- To remove the x and y terms in the general equation, translate the axes through the point (α, β), where α and β are the solutions to the equations:
- 18α + 24β - 6 = 0
- 24α + 4ẞ + 20 = 0
- After translation, the transformed equation will have no x and y terms.
Transforming to Remove xy and x, y Terms
- To transform an equation to remove both the xy term and the x, y terms, first translate the axes to remove the x, y terms, and then rotate the axes to remove the xy term.
- The angle of rotation is θ = (1/2) * tan⁻¹(2h/(a - b)).
General Strategy for Transforming Coordinates
- To remove the xy term: rotate the axes.
- To remove the x and y terms: translate the axes.
- To remove both xy, x, and y terms: perform both translation and rotation in the appropriate order.
Studying That Suits You
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.
Related Documents
Description
Explore the fundamental concepts of transformation of coordinates including translation and rotation of axes. Understand how changing the origin affects the coordinates of points and curves. This quiz will test your knowledge of the equations and principles governing these transformations.
