Transforming Coordinates and Rotating Axes

What is the polar coordinate for the point P(1, √3)?

The Cartesian coordinates can be directly converted to polar coordinates using the formula $r = rac{y}{x}$.

What is the value of θ for point P(1, √3) in degrees?

To convert Cartesian coordinates $(x,y)$ to polar coordinates, the formula for $r$ is $r = ext{(x, y)}$.

Match the following points to their respective coordinates:

What is the new equation of the curve after rotation if the original equation is $5x^2 + 2xy + 5y^2 = 2$?

The coefficient of $xy$ does not change when the axes are rotated.

What is the formula used to find the angle of rotation θ that eliminates the xy term from the equation?

The relationship between the old and new coordinates during rotation is defined by ______.

Match the following terms with their definitions:

Transforming Coordinates

Polar to Cartesian:
- Use the formulas:
  - x = r cos θ
  - y = r sin θ
Cartesian to Polar:
- Use the formulas:
  - r = √(x² + y²)
  - θ = tan⁻¹(y/x)
  - Be mindful of the quadrant of the point (x,y) to determine correct value of θ

Rotating Axes

Rotating the Coordinate Axes:
- The formulas for rotating the axes by an angle θ are:
  - x = X cos θ - Y sin θ
  - y = X sin θ + Y cos θ
  - Inverse equations:
    - X = x cos θ + y sin θ
    - Y = -x sin θ + y cos θ
- The coefficients of x and y are the transformation matrix:
  
  X Y
  
  x cos θ -sin θ
  
  y sin θ cos θ
Eliminating the xy Term:
- The coefficient of the xy term in the transformed equation is 2h(cos²θ - sin²θ) - 2(a - b) sin θ cos θ.
- To eliminate the xy term, set this coefficient to zero:
  - tan 2θ = 2h / (a - b)
  - θ = (1/2) tan⁻¹(2h / (a - b))
Transforming to Simplest Form:
- Step 1: Translate the axes to a new origin O'(α, β) by finding the values of α and β that make the partial derivatives of the original equation equal to 0.
- Step 2: Find the angle θ to eliminate the xy term in the transformed equation by using the formula θ = (1/2) tan⁻¹(2h / (a - b)).
- Step 3: Apply the rotation using the formulas x = X cos θ - Y sin θ and y = X sin θ + Y cos θ.
- Step 4: Substitute the translated and rotated coordinates into the original equation to obtain the simplest form.
Note: The sum of the coefficients of x² and y² (a + b) remains unchanged under rotation.

	X	Y
x	cos θ	-sin θ
y	sin θ	cos θ

This quiz covers the transformation of coordinates between polar and Cartesian systems, including the necessary formulas for conversion. Additionally, it delves into the rotation of axes and the associated transformation matrix. Test your understanding of these fundamental concepts in coordinate geometry.