Problem 62
Question
Write an equation of an ellipse in standard form with center at the origin and with the given characteristics. \(a=3, b=2,\) width 4
Step-by-Step Solution
Verified Answer
The equation of the ellipse with the given characteristics is \(\frac{x^2}{9} + \frac{y^2}{4} = 1\).
1Step 1: Write down the standard equation of an ellipse
The standard form of an equation of an ellipse with center at the origin is \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\), where \(a\) is the length of the semi-major axis and \(b\) is the length of the semi-minor axis.
2Step 2: Substitute the values of a and b into the equation
Given that \(a=3\) and \(b=2\), as provided in the problem, substitute these values into the standard equation of an ellipse. This results in \(\frac{x^2}{3^2}+\frac{y^2}{2^2}=1\).
3Step 3: Simplify the equation
Simplify the equation to \(\frac{x^2}{9} + \frac{y^2}{4} = 1\). This is the equation of the ellipse in standard form with center at the origin and the given characteristics.
Key Concepts
Standard Form of EllipseSemi-Major AxisSemi-Minor AxisCenter at the Origin
Standard Form of Ellipse
The standard form of the equation of an ellipse is a fundamental concept in geometry. It is presented as \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \). This equation is specific to ellipses whose center is located at the origin, which means the point (0,0). The variables \(x\) and \(y\) represent the coordinates of any point on the ellipse.
The square terms \(a^2\) and \(b^2\) in the denominators are crucial in defining the ellipse's shape and orientation. \(a\) represents the semi-major axis, while \(b\) represents the semi-minor axis. This mathematical expression is essential for identifying and describing the shape and dimensions of an ellipse in a Cartesian plane.
The square terms \(a^2\) and \(b^2\) in the denominators are crucial in defining the ellipse's shape and orientation. \(a\) represents the semi-major axis, while \(b\) represents the semi-minor axis. This mathematical expression is essential for identifying and describing the shape and dimensions of an ellipse in a Cartesian plane.
Semi-Major Axis
The semi-major axis of an ellipse is the longest radius that stretches from the center to the outer edge, running along the largest dimension of the ellipse. In our example, the semi-major axis is denoted by \(a\) and given as 3 units. This makes \(a^2 = 9\) within the standard form equation.
This axis plays a central role in determining the overall size of the ellipse. As it defines the longest span across the ellipse, the semi-major axis is typically aligned with either the x-axis or the y-axis, depending on the orientation of the ellipse.
This axis plays a central role in determining the overall size of the ellipse. As it defines the longest span across the ellipse, the semi-major axis is typically aligned with either the x-axis or the y-axis, depending on the orientation of the ellipse.
- Ellipses with their major axis aligned horizontally use the equation \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \).
- Ellipses with a vertical major axis use the equation \( \frac{x^2}{b^2} + \frac{y^2}{a^2} = 1 \).
Semi-Minor Axis
The semi-minor axis is the shorter radius of the ellipse, extending from the center to the edge perpendicular to the semi-major axis. In our example, the semi-minor axis (\(b\)) is 2 units long, thereby giving \(b^2 = 4\) in the ellipse equation.
This shorter axis determines the width of the ellipse. While it's always shorter than the semi-major axis, its length significantly shapes the overall appearance of the ellipse. The ellipse becomes more elongated when the difference between the semi-major and semi-minor axis lengths increases.
This shorter axis determines the width of the ellipse. While it's always shorter than the semi-major axis, its length significantly shapes the overall appearance of the ellipse. The ellipse becomes more elongated when the difference between the semi-major and semi-minor axis lengths increases.
- If \(b = a\), the ellipse is a circle, reflecting equal lengths of radii in all directions.
- The semi-minor axis is critical in calculations for any properties related to elliptical shapes, like area or eccentricity.
Center at the Origin
An ellipse centered at the origin means its geometric center is at the point (0,0) on a coordinate plane. This positioning simplifies the equation to the standard form, making calculations more straightforward as there are no linear shifts in either the x or y directions.
For ellipses not centered at the origin, you would see added terms \((x-h)^2\) and \((y-k)^2\) in the equation, with \(h\) and \(k\) being the x and y coordinates of the center respectively. However, with the origin as the center, these additional variables vanish. This aids immensely in solving problems involving symmetry and in comparing ellipses more easily.
For ellipses not centered at the origin, you would see added terms \((x-h)^2\) and \((y-k)^2\) in the equation, with \(h\) and \(k\) being the x and y coordinates of the center respectively. However, with the origin as the center, these additional variables vanish. This aids immensely in solving problems involving symmetry and in comparing ellipses more easily.
- An origin-centered ellipse is symmetric about both axes, providing a clean, even shape.
- It's the foundation for understanding more complex ellipses displaced from the origin.
Other exercises in this chapter
Problem 61
Find the center and the radius of each circle. $$ (x+3)^{2}+(y-5)^{2}=38 $$
View solution Problem 62
Which conic section is represented by the equation \(x^{2}+y^{2}=6 x-14 y-9 ?\) F. circle G. ellipse H. parabola J. hyperbola
View solution Problem 62
Suppose \(z\) varies jointly with \(x\) and \(y .\) Write a function that models each relationship. Find the value of \(z\) when \(x=-2\) and \(y=3 .\) \(z=32\)
View solution Problem 62
Solve each equation. $$ \log 8 x=3 $$
View solution