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x² – 9y2 : -9y2 = 13 x² + y² = 53 %3D

x² – 9y2 : -9y2 = 13 x² + y² = 53 %3D

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Solve conic equation

### System of Equations in Two Variables The image contains a system of two equations involving two variables, \(x\) and \(y\): 1. The first equation is: \[ x^2 - 9y^2 = 13 \] 2. The second equation is: \[ x^2 + y^2 = 53 \] This system can be analyzed and solved using various algebraic methods, such as substitution or elimination. These equations represent a hyperbola and a circle, respectively, in the coordinate plane. #### Equation 1: \( x^2 - 9y^2 = 13 \) This equation represents a hyperbola. The standard form of a hyperbola equation is \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \), and this particular hyperbola opens along the x-axis. #### Equation 2: \( x^2 + y^2 = 53 \) This equation represents a circle. The standard form of a circle equation is \( x^2 + y^2 = r^2 \), where \( r \) is the radius. In this case, the radius \( r \) is \(\sqrt{53}\). To find the points of intersection, if any, between the hyperbola and the circle, we would solve the system of equations simultaneously. ### Graphical Representation If you were to graph these equations: - The circle would be centered at the origin (0, 0) with a radius of approximately 7.28 units (\(\sqrt{53} \approx 7.28\)). - The hyperbola would also be centered at the origin, with its vertices aligned along the x-axis and the branches opening to the left and right. For educational purposes, it might be useful to plot these equations on a coordinate plane to visually inspect their points of intersection. Mathematically solving these equations will yield the exact points where the hyperbola and circle intersect.
Transcribed Image Text:### System of Equations in Two Variables The image contains a system of two equations involving two variables, \(x\) and \(y\): 1. The first equation is: \[ x^2 - 9y^2 = 13 \] 2. The second equation is: \[ x^2 + y^2 = 53 \] This system can be analyzed and solved using various algebraic methods, such as substitution or elimination. These equations represent a hyperbola and a circle, respectively, in the coordinate plane. #### Equation 1: \( x^2 - 9y^2 = 13 \) This equation represents a hyperbola. The standard form of a hyperbola equation is \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \), and this particular hyperbola opens along the x-axis. #### Equation 2: \( x^2 + y^2 = 53 \) This equation represents a circle. The standard form of a circle equation is \( x^2 + y^2 = r^2 \), where \( r \) is the radius. In this case, the radius \( r \) is \(\sqrt{53}\). To find the points of intersection, if any, between the hyperbola and the circle, we would solve the system of equations simultaneously. ### Graphical Representation If you were to graph these equations: - The circle would be centered at the origin (0, 0) with a radius of approximately 7.28 units (\(\sqrt{53} \approx 7.28\)). - The hyperbola would also be centered at the origin, with its vertices aligned along the x-axis and the branches opening to the left and right. For educational purposes, it might be useful to plot these equations on a coordinate plane to visually inspect their points of intersection. Mathematically solving these equations will yield the exact points where the hyperbola and circle intersect.
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