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Questions and Answers
What is the main goal in the examples from Section #3?
What is the main goal in the examples from Section #3?
- To translate the axis to eliminate first degree terms (correct)
- To calculate the slope of the lines
- To determine the intersection points of the curves
- To find the area of the curves
What does the angle θ represent in Section #4?
What does the angle θ represent in Section #4?
- The angle by which axes are rotated (correct)
- The distance between points
- The slope of the line
- The radius of curvature
In Example 1.3 from Section #3, which term needs to be eliminated?
In Example 1.3 from Section #3, which term needs to be eliminated?
- The xy mixed term
- The constant term
- The first degree term (correct)
- The second degree term
If the axes rotate by angle $ rac{eta}{4}$, what is the effect on the equation in Example 1.3 of Section #4?
If the axes rotate by angle $ rac{eta}{4}$, what is the effect on the equation in Example 1.3 of Section #4?
What is the reference equation for finding the new equations after rotation in Example 1.4?
What is the reference equation for finding the new equations after rotation in Example 1.4?
What is the relation shown in Example 1.1 involving the equation $(x^2 + y^2)^2 = x^2 - y^2$?
What is the relation shown in Example 1.1 involving the equation $(x^2 + y^2)^2 = x^2 - y^2$?
What does Example 1.5 illustrate about the point P moving with 3 units from the origin?
What does Example 1.5 illustrate about the point P moving with 3 units from the origin?
In Example 1.3, what does the equation $6y^2 - xy - x^2 + 30y + 36 = 0$ represent?
In Example 1.3, what does the equation $6y^2 - xy - x^2 + 30y + 36 = 0$ represent?
What type of calculation is performed in Example 1.2 to find the area of triangle ABC with vertices A(3, 2), B(−1, 4), and C(0, 3)?
What type of calculation is performed in Example 1.2 to find the area of triangle ABC with vertices A(3, 2), B(−1, 4), and C(0, 3)?
What is the significance of the ratio 2:1 mentioned in Example 1.6 concerning point Q related to the line 3x+y−1=0?
What is the significance of the ratio 2:1 mentioned in Example 1.6 concerning point Q related to the line 3x+y−1=0?
Flashcards
Cartesian Coordinates
Cartesian Coordinates
A coordinate system that uses two perpendicular axes (x and y) to locate points in a plane.
Polar Coordinates
Polar Coordinates
A coordinate system that uses a distance (r) from a fixed point (pole) and an angle (θ) from a fixed ray (polar axis) to locate points.
Area of a triangle
Area of a triangle
The formula to determine the area of a triangle given its vertices (x1, y1), (x2, y2), (x3, y3).
Equation of a pair of straight lines
Equation of a pair of straight lines
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Geometric Locus
Geometric Locus
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Rotating axes to eliminate xy term
Rotating axes to eliminate xy term
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Translation of axes to eliminate linear terms
Translation of axes to eliminate linear terms
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Transformation for rotated axes
Transformation for rotated axes
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New equation after translation and rotation
New equation after translation and rotation
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Equation of a line in rotated coordinate system
Equation of a line in rotated coordinate system
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Study Notes
Section One
- Example 1.1: Relates Cartesian (x, y) and polar (r, θ) coordinates for curves. The formula (x² + y²)² = x² - y² is given as an example.
- Example 1.2: Relates Cartesian and polar coordinates for curves. The formula r²(cos²(θ) - 2cos(θ)sin(θ) + 4sin²(θ)) = 1 is provided.
- Example 1.3: Establishes the relationship between Cartesian and polar coordinates of curves. Gives 9x² + 4y² = 36 as an example.
- Example 1.4: Connects Cartesian and polar coordinates for curves with the equation r² = 4 + 5cos(θ).
- Example 1.5: Determines the locus (geometric path) of a point 3 units from the origin.
- Example 1.6: A point P moves along the line 3x + y - 1 = 0. Point Q divides the distance OP in a 2:1 ratio internally. The locus of Q is sought.
Section Two
- Example 1.1: Proves that the area of a triangle with vertices (1,1), (m², m), and (n², n) is numerically equal to ½(1 - m)(m - n)(1 - n).
- Example 1.2: Calculates the area of a triangle ABC with vertices A(3, 2), B(-1, 4), and C(0, 3). Finds the length of the perpendicular from C to AB and the angle ∠ABC.
- Example 1.3: Demonstrates that the equation 6y² − xy − x² + 30y + 36 = 0 represents a pair of straight lines. Finds the angle between these lines and their intersection point.
Section Three
- Example 1.1: Finds a new origin (a, β) to eliminate first-degree terms from the equation x² + y² – 4x – 6y + 4 = 0 when translating axes.
- Example 1.2: Finds a new origin (a, β) to eliminate first-degree terms from the equation 3x² − 2xy + 4y² − 3x − 10y − 7 = 0 by translating axis.
- Example 1.3: Finds a new origin (a, β) to eliminate first-degree terms from the equation x² + 4xy + 4y² + 8y + 11 = 0 by translating axes.
Section Four
- Example 1.1: Determines the angle θ for rotating axes to eliminate the xy term from the equation x² - 2xy + y² = 4.
- Example 1.2: Finds the angle θ for rotating axes to eliminate the xy term from equation x² – 2xy + y² = 4.
- Example 1.3: If the origin is moved to (−1, 2) and the axes rotated by an angle, find the new equation of the curve 4x² + y² + 8x − 4y + 7 = 0.
- Example 1.4: Finds the new equations for a line and a curve x² − 4xy + y² = 1 when the axes rotate by an angle without changing the origin.
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Description
Test your knowledge on the relationships between Cartesian and polar coordinates in curves. This quiz includes various examples and calculations related to geometric paths and areas in coordinate geometry. Ideal for students studying coordinate geometry concepts.
