Problem 81
Question
Exercises \(81-83\) will help you prepare for the material covered in the first section of the next chapter. Consider the equation \(\frac{x^{2}}{9}+\frac{y^{2}}{4}=1\) a. Set \(y=0\) and find the \(x\) -intercepts. b. Set \(x=0\) and find the \(y\) -intercepts.
Step-by-Step Solution
Verified Answer
The x-intercepts are \(±3\), and the y-intercepts are \(±2\).
1Step 1: Find the x-intercepts
To find the x-intercepts, set \(y=0\) in the given equation.This is achieved by inserting \(y=0\) into \(\frac{x^{2}}{9}+\frac{y^{2}}{4}=1\). The equation simplifies to \(\frac{x^{2}}{9}+0=1\), which simplifies further to \(\frac{x^{2}}{9}=1\). By multiplying both sides by 9, the equation \(x^{2}=9\) is obtained. Solving for \(x\), it is found that \(x = ± \sqrt {9}\), which simplifies to the answer: \(x=±3\).
2Step 2: Find the y-intercepts
To find the y-intercepts, place \(x=0\) into the equation. Inserting \(x=0\) into \(\frac{x^{2}}{9}+\frac{y^{2}}{4}=1\) results in the equation \(\frac{0}{9}+\frac{y^{2}}{4}=1\), which simplifies to \(\frac{y^{2}}{4}=1\). Multiplying both sides by 4, the equation \(y^{2} = 4\) is obtained. Solving for \(y\), it can be found that \(y = ±\sqrt {4}\), which simplifies to the following answer: \(y=±2\).
Key Concepts
Equation of EllipseSolving Quadratic EquationsGraphing EllipsesCoordinate Geometry
Equation of Ellipse
In coordinate geometry, the equation of an ellipse is a way to express the relationship between the horizontal (x) and vertical (y) distances of any point from the center of the ellipse. The standard form of an ellipse's equation is \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), where \( a \) and \( b \) are the lengths of the semi-major and semi-minor axes, respectively.
An ellipse is basically a stretched circle with two axes of symmetry. The larger axis is called the major axis, while the shorter one is termed the minor axis. If an ellipse is rotated so that these axes align with the x and y axes of the coordinate plane, the equation becomes easier to work with, as in the given exercise.
When identifying x and y intercepts of an ellipse, you leverage the understanding that intercepts lie on the coordinate axes, simplifying the equation by setting the other variable to zero, as shown in the original solution.
An ellipse is basically a stretched circle with two axes of symmetry. The larger axis is called the major axis, while the shorter one is termed the minor axis. If an ellipse is rotated so that these axes align with the x and y axes of the coordinate plane, the equation becomes easier to work with, as in the given exercise.
When identifying x and y intercepts of an ellipse, you leverage the understanding that intercepts lie on the coordinate axes, simplifying the equation by setting the other variable to zero, as shown in the original solution.
Solving Quadratic Equations
Quadratic equations are a fundamental part of algebra and can be recognized by their highest degree term, which is squared (i.e., \( x^2 \)). The general form of a quadratic equation is \( ax^2 + bx + c = 0 \), where \( a \) is not equal to zero.
There are several methods for solving quadratic equations, such as factoring, completing the square, using the quadratic formula, and graphing. In the context of finding intercepts for an ellipse, you only need to solve a simple form where the equation is already set equal to a constant. By isolating the squared term and taking its square root, as seen in the step-by-step solution for x and y intercepts, we find the possible values of the variable when the other is set to zero.
There are several methods for solving quadratic equations, such as factoring, completing the square, using the quadratic formula, and graphing. In the context of finding intercepts for an ellipse, you only need to solve a simple form where the equation is already set equal to a constant. By isolating the squared term and taking its square root, as seen in the step-by-step solution for x and y intercepts, we find the possible values of the variable when the other is set to zero.
Graphing Ellipses
Graphing ellipses involves plotting points on a coordinate system that satisfies the equation of an ellipse. Once you determine the lengths of the semi-major and semi-minor axes from the equation, these measurements can be used to draw the shape of the ellipse on a grid.
To graph an ellipse accurately, it's crucial to identify the center of the ellipse, often at the origin in its standard form, and then plot the vertices and co-vertices which lie on the axes. The x and y intercepts found in the original solution help to locate these crucial points. The vertices are at the ends of the major axis, while the co-vertices are at the ends of the minor axis. For the given ellipse equation, the points (±3, 0) and (0, ±2) are the vertices and co-vertices, respectively. After plotting these points, you can sketch the curve of the ellipse, making sure it touches each of these points.
To graph an ellipse accurately, it's crucial to identify the center of the ellipse, often at the origin in its standard form, and then plot the vertices and co-vertices which lie on the axes. The x and y intercepts found in the original solution help to locate these crucial points. The vertices are at the ends of the major axis, while the co-vertices are at the ends of the minor axis. For the given ellipse equation, the points (±3, 0) and (0, ±2) are the vertices and co-vertices, respectively. After plotting these points, you can sketch the curve of the ellipse, making sure it touches each of these points.
Coordinate Geometry
Coordinate geometry, also known as analytic geometry, uses algebraic equations to describe geometric shapes on a coordinate plane. This method provides a bridge between algebra and geometry, allowing for a visual representation of algebraic equations.
The coordinate plane consists of two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical). When dealing with shapes like circles, parabolas, or ellipses, coordinate geometry aids in graphing these shapes and deriving properties such as intercepts, area, and eccentricity.
Intercepts are points where the graph of an equation crosses the axes. As you've seen in the example solved above, finding the x and y intercepts is a common task, critical for graphing and understanding the dimensions and orientation of shapes within the coordinate plane.
The coordinate plane consists of two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical). When dealing with shapes like circles, parabolas, or ellipses, coordinate geometry aids in graphing these shapes and deriving properties such as intercepts, area, and eccentricity.
Intercepts are points where the graph of an equation crosses the axes. As you've seen in the example solved above, finding the x and y intercepts is a common task, critical for graphing and understanding the dimensions and orientation of shapes within the coordinate plane.
Other exercises in this chapter
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