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Chapter 9: Problem 38

Find an equation of the ellipse that satisfies the given conditions. \(x\) -intercepts \(\pm 3, y\) -intercepts \(\pm \frac{1}{2}\)

Short Answer

Expert verified
The equation of the ellipse is \(\frac{x^2}{9} + 4y^2 = 1\).

Step by step solution

01

Identify the x-intercepts and y-intercepts

The ellipse has x-intercepts at \(\pm 3\) and y-intercepts at \(\pm \frac{1}{2}\). The distance from the origin to the x-intercepts is the semi-major axis, denoted by \(a\), and the distance from the origin to the y-intercepts is the semi-minor axis, denoted by \(b\).
02

Determine the values of \(a\) and \(b\)

The lengths of the semi-major and semi-minor axes are the distances from the center of the ellipse to the intercepts. So, \[a = \frac{1}{2}(\text{distance from -3 to 3}) = \frac{1}{2}(6) = 3\] \[b = \frac{1}{2}(\text{distance from -\frac{1}{2} to \frac{1}{2}}) = \frac{1}{2}(1) = \frac{1}{2}\]
03

Write the equation of the ellipse

Now that we have the values of \(a\) and \(b\), we can write the equation of the ellipse in standard form. Since the ellipse is centered at the origin, the equation takes the form: \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\] Plugging in the values of \(a\) and \(b\) we just found: \[\frac{x^2}{3^2} + \frac{y^2}{\left( \frac{1}{2} \right)^2} = 1\]
04

Simplify the equation

Simplify the equation to obtain the final result: \[\frac{x^2}{9} + \frac{y^2}{\frac{1}{4}} = 1\] \[\frac{x^2}{9} + 4y^2 = 1\] Therefore, the equation of the ellipse is: \[\frac{x^2}{9} + 4y^2 = 1\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Conic Sections
Conic sections are curves obtained by intersecting a plane with a double-napped cone. Every shape such as ellipse, circle, parabola, and hyperbola falls under conic sections. Ellipses are set apart by their unique shape, resembling an elongated circle.

They are among the most common shapes in geometry and have various real-world applications, from planetary orbits to lenses. Understanding conic sections helps in grasping the basic geometric principles behind many natural phenomena. The position and intersection angle of the plane with the cone determine the type of conic section formed.
X-intercepts
X-intercepts are points where a curve crosses the x-axis of a graph. In the context of an ellipse, these are the points where the ellipse meets the x-axis. For our ellipse, the x-intercepts are \(\pm 3\).

This means the ellipse crosses the x-axis at both -3 and 3. These intercepts help in determining the concept of semi-major and semi-minor axes, which are crucial for defining the shape of the ellipse. Understanding x-intercepts aids in analyzing the symmetry and periodic behavior of the graph on a coordinate plane.
Y-intercepts
Y-intercepts occur at the points where a curve intersects the y-axis of a graph. In the given ellipse, the y-intercepts are \(\pm \frac{1}{2}\), signifying points of intersection at both -\(\frac{1}{2}\) and \(\frac{1}{2}\) along the y-axis.

Just like x-intercepts, these are instrumental in pinpointing the semi-minor and semi-major axes of the ellipse. Understanding the y-intercepts helps in determining the vertical symmetry of the ellipse and is key for drafting the complete graph in 2D spaces.
Semi-major Axis
The semi-major axis is the longest radius of an ellipse. In the equation of an ellipse, it is represented as \(a\). For our specific equation, the semi-major axis is defined by the distance of the x-intercepts: \[a = 3\].

This value is crucial because it determines the length one must travel from the center of the ellipse along the major axis to reach the edge of the curve. It helps in defining the shape and the spread of the ellipse on a graph, offering insights into its dimensional properties.
Semi-minor Axis
The semi-minor axis is the shortest radius of an ellipse. It is depicted as \(b\) in the ellipse's equation. For this ellipse, it's defined by the distance of the y-intercepts: \[b = \frac{1}{2}\].

The semi-minor axis determines how stretched or compressed an ellipse is in the direction perpendicular to the semi-major axis. Understanding the semi-minor axis is vital for fully grasping the proportions and overall area of the ellipse, providing key insights into its geometric properties.

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Most popular questions from this chapter

Find parametric equations for the Folium of Descartes, \(x^{3}+y^{3}=3 a x y\) with parameter \(t=y / x\).

Write a polar equation of the conic that has a focus at the origin and the given properties. Identify the conic. Eccentricity 1, directrix \(y=-3\)

Let \(P\) be a point other than the origin lying on the curve \(r=f(\theta)\). If \(\psi\) is the angle between the tangent line to the curve at \(P\) and the radial line \(O P\), then \(\tan \psi=\frac{r}{d r / d \theta} .\) (See Section 9.4, Exercise 84.) a. Show that the angle between the tangent line to the loga rithmic spiral \(r=e^{m \theta}\) and the radial line at the point of tangency is a constant. b. Suppose the curve with polar equation \(r=f(\theta)\) has the property that at any point on the curve, the angle \(\psi\) between the tangent line to the curve at that point and the radial line from the origin to that point is a constant. Show that \(f(\theta)=C e^{m \theta}\), where \(C\) and \(m\) are constants.

Show that the parabolas with polar equations $$ r=\frac{c}{1+\sin \theta} \quad \text { and } \quad r=\frac{d}{1-\sin \theta} $$ intersect at right angles.

The cornu spiral is a curve defined by the parametric equations \(x=C(t)=\int_{0}^{t} \cos \left(\pi u^{2} / 2\right) d u \quad y=S(t)=\int_{0}^{t} \sin \left(\pi u^{2} / 2\right) d u\) where \(C\) and \(S\) are called Fresnel integrals. They are used to explain the phenomenon of light diffraction. a. Plot the spiral. Describe the behavior of the curve as \(t \rightarrow \infty\) and as \(t \rightarrow-\infty\). b. Find the length of the spiral from \(t=0\) to \(t=a\).
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