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Questions and Answers
What is the polar coordinate for the point P(1, √3)?
What is the polar coordinate for the point P(1, √3)?
- (1, 60°)
- (2, 90°)
- (√4, 30°)
- (2, 60°) (correct)
The Cartesian coordinates can be directly converted to polar coordinates using the formula $r = rac{y}{x}$.
The Cartesian coordinates can be directly converted to polar coordinates using the formula $r = rac{y}{x}$.
False (B)
What is the value of θ for point P(1, √3) in degrees?
What is the value of θ for point P(1, √3) in degrees?
60°
To convert Cartesian coordinates $(x,y)$ to polar coordinates, the formula for $r$ is $r = ext{(x, y)}$.
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:
Match the following points to their respective coordinates:
What is the new equation of the curve after rotation if the original equation is $5x^2 + 2xy + 5y^2 = 2$?
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.
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?
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 ______.
The relationship between the old and new coordinates during rotation is defined by ______.
Match the following terms with their definitions:
Match the following terms with their definitions:
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Study Notes
Transforming Coordinates
- Polar to Cartesian:
- Use the formulas:
- x = r cos θ
- y = r sin θ
- Use the formulas:
- 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
- Use the formulas:
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
- The formulas for rotating the axes by an angle θ are:
- 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.
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