Transformation of Coordinates Concepts
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Transformation of Coordinates Concepts

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Questions and Answers

What occurs to the coordinates of a point when a new set of axes is taken?

  • They remain unchanged.
  • They become complex numbers.
  • They change according to the new axes. (correct)
  • They only change if the point moves.

    • What is the purpose of translating the origin of coordinates?

  • To keep the direction of the axes constant. (correct)
  • To change the direction of axes.
  • To rotate the axes.
  • To change the coordinates of all points.

    • How do the equations of transformation relate to the coordinates of a point P?

  • They relate only to the new axes.
  • They relate to both sets of axes with constant values.
  • They are independent of the coordinates of O'.
  • They involve adding the coordinates of the origin. (correct)

    • What mathematical relationships are used for rotation of axes?

    <p>$x = x' cos θ + y' sin θ$ and $y = -x' sin θ + y' cos θ$</p> Signup and view all the answers

    What is a key property of coordinates when transforming to a new set of axes?

    <p>The equation of the curve changes.</p> Signup and view all the answers

    In the transformation of coordinates, what does the variable θ represent?

    <p>The angle of rotation.</p> Signup and view all the answers

    Which equation represents the transformation from old to new axes during translation?

    <p>$x' = x - α$ and $y' = y - β$</p> Signup and view all the answers

    What geometric shape does the transformation of coordinates imply?

    <p>A linear transformation.</p> Signup and view all the answers

    What is the correct expression for the transformed equation after substitution?

    <p>a'x² + 2h'xy + b'y² + 2g'x + 2f'y + c' = 0</p> Signup and view all the answers

    Which is the correct relationship derived from the transformed coefficients?

    <p>a'b' - h'² = ab - h²</p> Signup and view all the answers

    What condition must be satisfied for the term xy to be absent in the transformed equation?

    <p>h' = 0</p> Signup and view all the answers

    What does the expression Δ denote in terms of coefficients?

    <p>abc + 2fgh - af² - bg² - ch²</p> Signup and view all the answers

    What transformation is applied to the variables x and y?

    <p>x = x' cos θ - y' sin θ, y = x' sin θ + y' cos θ</p> Signup and view all the answers

    What is the formula used to find the angle θ for the rotation of axes?

    <p>θ = 1/2 * tan^{-1}(2h/(a-b))</p> Signup and view all the answers

    What invariant remains unchanged under the transformation of the equation?

    <p>a'b' - h^2</p> Signup and view all the answers

    During the transformation, how is the coefficient g' expressed?

    <p>g' = g cos θ + f sin θ</p> Signup and view all the answers

    When the original equation 5x² + 4xy + 5y² - 10 = 0 is transformed, which term is removed?

    <p>xy</p> Signup and view all the answers

    What is the significance of the equation a'b' - h'² = ab - h²?

    <p>It relates the original and transformed coefficients.</p> Signup and view all the answers

    How does the expression for a' involve the trigonometric function?

    <p>a' = a cos²θ + b sin²θ</p> Signup and view all the answers

    What values are assigned to a, b, and h in the problem to verify the transformation?

    <p>a = 5, b = 5, h = 2</p> Signup and view all the answers

    Which equation derives from substituting the rotated coordinates in the original equation?

    <p>5(x² - 2xy + y²) + 4(x² - y²) + 5(x² + 2xy + y²) - 20 = 0</p> Signup and view all the answers

    What is the result of comparing the coefficients in the transformed equation?

    <p>It retains the relationship a' + b' = a + b.</p> Signup and view all the answers

    What characteristic does the quantity A represent in the context of transformations?

    <p>A represents the invariants of transformation.</p> Signup and view all the answers

    What substitution is used for x in the transformation process?

    <p>x = x cos π/4 - y sin π/4</p> Signup and view all the answers

    What is the expression for $x'$ in terms of $x$ and $y$ during the transformations?

    <p>$x' = x cos θ + y sin θ$</p> Signup and view all the answers

    In the transformation to axes inclined at 45°, what are the values of $cos θ$ and $sin θ$?

    <p>$cos θ = 1/√2$ and $sin θ = 1/√2$</p> 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?

    <p>It creates a transformation of the equation with unchanged highest degree terms.</p> Signup and view all the answers

    How is $y'$ expressed in terms of $x$ and $y$ during the transformations?

    <p>$y' = -x sin θ + y cos θ$</p> Signup and view all the answers

    What remains unchanged in the transformed equation of a curve according to the transformations described?

    <p>The coefficients of the highest degree terms remain unchanged.</p> Signup and view all the answers

    When transitioning to axes inclined at an angle, what general structure does the transformation follow?

    <p>The scheme can be interpreted both horizontally or vertically.</p> 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?

    <p>New transformation equations combine effects of both shifts and rotations.</p> Signup and view all the answers

    What can be concluded about the constant term after substituting $α$ and $β$ into the general equation?

    <p>It is derived from substituting $α$ and $β$ into the original expression.</p> Signup and view all the answers

    What is the transformed equation of 9x² + 24xy + 2y² - 6x + 20y + 41 = 0 through the point (-1, 1)?

    <p>9x² + 24xy + 2y² + 54 = 0</p> Signup and view all the answers

    Which condition must be satisfied to remove the terms in x and y from the transformed equation?

    <p>18α + 24β - 6 = 0 and 24α + 43 + 20 = 0</p> Signup and view all the answers

    What is the value of θ when the term in xy is removed by rotating the axes?

    <p>tan⁻¹(3/4)</p> Signup and view all the answers

    What is the correct substitution for x when performing the transformation for rotating axes?

    <p>x cos θ - y sin θ</p> Signup and view all the answers

    Which transformation is needed to remove the term of y in the equation 3x - 25y + 41 = 0?

    <p>Parallel axes through (-3, 2)</p> Signup and view all the answers

    What manual calculation is required for the transformation of y(y - 2a) = ax?

    <p>Parallel axes through (-a, a)</p> Signup and view all the answers

    What is the resulting form of the equation 3x² + 5y² - 3 = 0 after rotating through 45°?

    <p>Transformed equation with xy terms present</p> Signup and view all the answers

    What determines the angle of inclination when transforming axes through the point (a, b)?

    <p>tan⁻¹(b/a)</p> 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.

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    Transformation of Coordinates PDF

    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.

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