Ch. 7 - Conic Sections

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook

Blitzer 8th Edition

Ch. 7 - Conic Sections

Problem 49

Chapter 8, Problem 49

Graph each ellipse and give the location of its foci. 9(x − 1)²+4(y+3)² = 36

Verified step by step guidance

Rewrite the given equation in standard form by dividing both sides by 36: $\frac{9(x - 1)^2}{36} + \frac{4(y + 3)^2}{36} = \frac{36}{36}$.

Simplify the fractions to get: $\frac{(x - 1)^2}{4} + \frac{(y + 3)^2}{9} = 1$.

Identify the values of $a^2$ and $b^2$ from the denominators. Here, $a^2 = 9$ and $b^2 = 4$. Since $a^2 > b^2$, the major axis is vertical.

Find the center of the ellipse, which is at the point $(h, k) = (1, -3)$ from the terms $(x - h)^2$ and $(y - k)^2$.

Calculate the focal distance $c$ using the formula $c^2 = a^2 - b^2$. Then, determine the coordinates of the foci by moving $c$ units along the major axis (vertical) from the center.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Standard Form of an Ellipse

An ellipse equation in standard form is written as (x-h)²/a² + (y-k)²/b² = 1, where (h, k) is the center. The values a² and b² represent the squares of the semi-major and semi-minor axes, respectively. Converting the given equation to this form is essential for graphing and analyzing the ellipse.

Identifying the Major and Minor Axes

The major axis is the longest diameter of the ellipse, and the minor axis is the shortest. By comparing a² and b², you determine which axis is major (larger value) and which is minor (smaller value). This helps in correctly sketching the ellipse and locating its vertices.

Foci of an Ellipse

The foci are two fixed points inside the ellipse such that the sum of distances from any point on the ellipse to the foci is constant. Their locations are found using c² = |a² - b²|, where c is the distance from the center to each focus along the major axis. Knowing the foci is crucial for understanding ellipse properties.

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Foci and Vertices of an Ellipse

Related Practice

Textbook Question

Convert each equation to standard form by completing the square on x and y. Then graph the ellipse and give the location of its foci. 9x² +25y² - 36x + 50y – 164 = 0

Textbook Question

Identify each equation without completing the square.

$y^2-4x-4y=0$

Textbook Question

Convert each equation to standard form by completing the square on x or y. Then find the vertex, focus, and directrix of the parabola. Finally, graph the parabola. x² + 6x - 4y + 1 = 0

Textbook Question

In Exercises 43–50, convert each equation to standard form by completing the square on x and y. Then graph the hyperbola. Locate the foci and find the equations of the asymptotes. $4 x^{2} - 25 y^{2} - 32 x + 164 = 0$

Textbook Question

Identify each equation without completing the square. y² - 4x + 2y + 21 = 0

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Textbook Question

In Exercises 51–56, graph each relation. Use the relation's graph to determine its domain and range. $\frac{x^{2}}{9} - \frac{y^{2}}{16} = 1$