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Questions and Answers
What type of conic section does the equation $13x^2 - 10xy + 13y^2 = 288$ represent?
What type of conic section does the equation $13x^2 - 10xy + 13y^2 = 288$ represent?
Which term in the equation $13x^2 - 10xy + 13y^2 = 288$ indicates it is not a simple quadratic equation?
Which term in the equation $13x^2 - 10xy + 13y^2 = 288$ indicates it is not a simple quadratic equation?
What is the significance of the coefficients in the equation $13x^2 - 10xy + 13y^2 = 288$?
What is the significance of the coefficients in the equation $13x^2 - 10xy + 13y^2 = 288$?
When rewriting the equation $13x^2 - 10xy + 13y^2 = 288$ in standard form, which step is essential?
When rewriting the equation $13x^2 - 10xy + 13y^2 = 288$ in standard form, which step is essential?
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What geometric properties can be derived from analyzing the equation $13x^2 - 10xy + 13y^2 = 288$?
What geometric properties can be derived from analyzing the equation $13x^2 - 10xy + 13y^2 = 288$?
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Study Notes
Conic Section Type
- The equation $13x^2 - 10xy + 13y^2 = 288$ represents an ellipse.
Non-Simple Quadratic Term
- The term -10xy indicates that the equation is not a simple quadratic equation. This term represents a cross-product between the x and y variables, which is characteristic of rotated conic sections.
Coefficient Significance
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The coefficients of the squared terms ($13x^2$ and $13y^2$) indicate the orientation of the ellipse. Since the coefficients are equal, the ellipse is centered at the origin and has a horizontal and vertical major and minor axis, respectively.
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The coefficient of the cross-product term (-10xy) determines the rotation of the ellipse relative to the standard coordinate axes. This rotation can be calculated using the formula θ = (1/2)arctan(-10/(13-13))
Standard Form Steps
- To rewrite the equation in standard form, rotating the coordinate axes is essential. This involves finding the rotation angle θ and transforming the equation into a new coordinate system (x', y') aligned with the ellipse's axes.
Ellipse Properties
- By analyzing the standard form equation, we can determine the center, major and minor axes, foci, eccentricity, and directrices of the ellipse.
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Description
This quiz explores the characteristics of the conic section represented by the equation 13x² - 10xy + 13y² = 288. It discusses the significance of coefficients, steps to rewrite the equation in standard form, and the geometric properties that can be derived from it. Test your understanding of conic sections and their properties.
