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Chapter 10: Problem 34

Sketch the graph of each ellipse and identify the foci. $$9(x-1)^{2}+4(y+3)^{2}=36$$

Short Answer

Expert verified
Center: (1, -3); Foci: (1, -3 卤 鈭�5)

Step by step solution

01

Rewrite the Equation in Standard Form

Divide the entire equation by 36 to normalize it: \[ \frac{9(x-1)^{2}}{36} + \frac{4(y+3)^{2}}{36} = 1 \] which simplifies to \[ \frac{(x-1)^{2}}{4} + \frac{(y+3)^{2}}{9} = 1 \].
02

Identify the Center, Major and Minor Axes

In the standard form, \(\frac{(x-h)^{2}}{a^{2}} + \frac{(y-k)^{2}}{b^{2}} = 1\), the center \((h, k)\) is at \((1, -3)\). Since \( b^{2} = 9 \) and \( a^{2} = 4 \), the lengths of the semi-major and semi-minor axes are \( b = 3 \) and \( a = 2 \), respectively. The major axis is vertical because 9 > 4.
03

Calculate the Distance to the Foci

The distance from the center to each focus \( c \) is found using \( c = \sqrt{b^{2} - a^{2}} \). Therefore, \[ c = \sqrt{9 - 4} = \sqrt{5} \].
04

Locate the Foci

Since the major axis is vertical, the foci are located at \[ (1, -3 \pm \sqrt{5}). \]
05

Sketch the Ellipse

Plot the center at \((1, -3)\). From the center, move 3 units up and down to plot the vertices, and 2 units left and right to plot the co-vertices. Draw a smooth ellipse passing through these points. Indicate the foci at \[ (1, -3 + \sqrt{5}) \text{ and } (1, -3 - \sqrt{5}). \]

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Key Concepts

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

Ellipse Standard Form
An ellipse is a rounded shape that looks like a stretched circle. The equation for an ellipse can be written in a special format called the standard form. This standard form is \( \frac{(x-h)^{2}}{a^{2}} + \frac{(y-k)^{2}}{b^{2}} = 1 \), where \

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Most popular questions from this chapter

Express \(\frac{2}{x+5}+\frac{x}{x-5}-\frac{3}{x^{2}}\) as a single rational expression.

Write each of the following equations in one of the forms: \(y=a(x-h)^{2}+k, \quad x=a(y-h)^{2}+k\) \(\frac{(x-h)^{2}}{a^{2}}+\frac{(y-k)^{2}}{b^{2}}=1,\) or \((x-h)^{2}+(y-k)^{2}=r^{2}\). Then identify each equation as the equation of a parabola, an ellipse, or a circle. $$2 x^{2}+4 y^{2}=4-y$$

Use rotation of axes to eliminate the product term and identify the type of conic. $$3 x^{2}-2 \sqrt{3} x y+y^{2}+(\sqrt{3}-1) x-(1+\sqrt{3}) y-1=0$$

Solve the system \(5 x+7 y=20, x-3 y+z=-10\) and \(2 y+5 z=50\)

Find the vertex, focus, and directrix of each parabola without completing the square, and determine whether the parabola opens upward or downward. $$y=\frac{1}{4} x^{2}+5$$
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