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Chapter 9: Problem 17

Find the foci and vertices of the ellipse, and sketch its graph. $$ 4 x^{2}+9 y^{2}=36 $$

Short Answer

Expert verified
The foci of the ellipse are at \((\pm\sqrt{5}, 0)\), and the vertices are at \((\pm 3, 0)\) along the major axis and \((0, \pm 2)\) along the minor axis. To sketch the graph, plot the center at the origin, foci on the major axis, and vertices on both the major and minor axes. Connect the vertices with a smooth curve that passes through the vertices and gets closer to the foci as it approaches the major axis.

Step by step solution

01

Rewrite the equation in standard form

Divide the given equation by 36 to get the ellipse in the standard form: \[ \frac{x^2}{9} + \frac{y^2}{4} = 1. \]
02

Identify values of semi-major and semi-minor axes

From the standard equation \(\frac{x^2}{9} + \frac{y^2}{4} = 1\), we can see that the length of the major and minor axes are given by: Major axis: \(2a = \sqrt{9} \times 2 = 6\) Minor axis: \(2b = \sqrt{4} \times 2 = 4\) Thus, a = 3 and b = 2.
03

Find the distance between the foci

To find the distance between the foci, we need to calculate the value of c using the relationship between the semi-major axis (a), the semi-minor axis (b), and the distance between the foci (c): \[ c = \sqrt{a^2 - b^2} = \sqrt{3^2 - 2^2} = \sqrt{9-4} = \sqrt{5} \] The foci are located \(c = \sqrt{5}\) units from the center of the ellipse along the major axis.
04

Identify foci and vertices

Using the values of a, b, and c, we can determine the foci and vertices of the ellipse: Foci: \((\pm\sqrt{5}, 0)\) Vertices: \((\pm 3, 0)\) along the major axis, \((0, \pm 2)\) along the minor axis.
05

Sketch the ellipse

To sketch the graph of the ellipse, use the following steps: 1. Draw the major and minor axes on the coordinate plane. 2. Plot the center of the ellipse at the origin (0, 0). 3. Plot the foci at points \((\pm\sqrt{5}, 0)\) on the major axis. 4. Plot the vertices at points \(\pm 3, 0\) on the major axis, and \((0, \pm2)\) on the minor axis. 5. Draw the ellipse by connecting the vertices with a smooth curve, making sure it passes through the vertices and gets closer to the foci as it approaches the major axis. By following these steps, we now have the graph of the ellipse \(\frac{x^2}{9} + \frac{y^2}{4} = 1\), with the foci \((\pm\sqrt{5}, 0)\), vertices \((\pm 3, 0) \) along the major axis and \((0, \pm 2)\) along the minor axis.

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Key Concepts

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

Standard Form of an Ellipse
The standard form of an ellipse equation is essential in understanding its geometric properties. When dealing with ellipses in algebra, we commonly encounter equations in the form \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), where \(a\) and \(b\) are the lengths of the semi-major and semi-minor axes, respectively. If \(a\) is greater than \(b\), the ellipse is stretched more along the x-axis; otherwise, it's stretched more along the y-axis if \(b > a\).
Ellipse Foci and Vertices
Understanding the foci and vertices gives insights into the ellipse鈥檚 shape and orientation. The vertices are the points where the ellipse intersects the major axis, while the foci are internal points \(c\) units away from the center, which have the unique property that the sum of the distances from any point on the ellipse to the foci is constant. This distance \(c\) can be found using the equation \(c = \sqrt{a^2 - b^2}\). In an ellipse with horizontal major axis, the vertices are at \(\pm a, 0\) and the foci at \(\pm c, 0\); with vertical major axis, the vertices are at \(0, \pm a\) and the foci at \(0, \pm c\).
Plotting Ellipses
Plotting an ellipse may seem daunting, but it鈥檚 a systematic process. Understand that the center of an ellipse is the midpoint of its major and minor axes. Always start by finding and marking the center on a graph. Next, plot the vertices along the major axis and co-vertices along the minor axis. The foci should be plotted at \(\pm c\) units from the center along the major axis. Finally, draw a smooth oval shape to connect these points, ensuring that it passes through each vertex and approaches near the foci. This practice emphasizes the symmetry of an ellipse about both axes.
Conic Sections in Calculus
In calculus, conic sections like ellipses are studied to explore the intersections of a plane and a double-napped cone. By altering the angle of intersection, we can derive different conic sections including circles, ellipses, parabolas, and hyperbolas. Calculations involving ellipses may include finding areas, perimeters (circumferences), and arc lengths using integral calculus. The fundamental property that the sum of the distances from any point on the ellipse to its foci is constant is frequently utilized in problem-solving scenarios involving ellipses.

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Most popular questions from this chapter

Let \(\psi\) be the angle between the radial line \(O P\) and the tangent line to the curve with polar equation \(r=f(\theta)\) at \(P\) (see the figure). Show that $$ \tan \psi=r \frac{d \theta}{d r} $$ Hint: Observe that \(\psi=\phi-\theta\). Then use the trigonometric identity $$ \tan (a-b)=\frac{\tan a-\tan b}{1+\tan a \tan b} $$

Sketch the curve with the polar equation. \(r=e^{\theta}, \quad \theta \geq 0 \quad\) (logarithmic spiral)

Show that a conic with focus at the origin, eccentricity \(e\), and directrix \(x=d\) has polar equation $$ r=\frac{e d}{1+e \cos \theta} $$

a. Plot the graphs of the cardioids \(r=a(1+\cos \theta)\) and \(r=a(1-\cos \theta)\). b. Show that the cardioids intersect at right angles except at the pole.

Find the slope of the tangent line to the curve at the point corresponding to the value of the parameter. $$ x=e^{2 t}, \quad y=\ln t ; \quad t=1 $$
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