Ch. 7 - Conic Sections

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Non è quello che usi tu?Cambia libro di testo

Blitzer 8th Edition

Ch. 7 - Conic Sections

Problema 87

Capitolo 8, Problema 87

Find the standard form of the equation of an ellipse with vertices at (0, -6) and (0, 6), passing through (2, 4).

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Identify the center of the ellipse by finding the midpoint of the vertices. Since the vertices are at (0, -6) and (0, 6), the center is at (0, 0).

Determine the orientation of the ellipse. The vertices lie on the y-axis, so the major axis is vertical. This means the standard form of the ellipse equation is $ \frac{x^2}{b^2} + \frac{y^2}{a^2} = 1 $, where $a$ is the distance from the center to a vertex along the y-axis.

Calculate $a$, the distance from the center to a vertex. Since the vertices are at (0, ±6), $a = 6$. So, $a^2 = 36$.

Use the point (2, 4) that lies on the ellipse to find $b^2$. Substitute $x = 2$, $y = 4$, and $a^2 = 36$ into the ellipse equation: $ \frac{2^2}{b^2} + \frac{4^2}{36} = 1 $.

Solve the equation from step 4 for $b^2$ by isolating $ \frac{4}{b^2} $, then multiply both sides by $b^2$ and solve for $b^2$. This will give you the value needed to write the standard form of the ellipse.

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Standard Form of an Ellipse

The standard form of an ellipse equation depends on the orientation of its major axis. For a vertical major axis centered at the origin, the equation is (x^2 / b^2) + (y^2 / a^2) = 1, where 'a' is the distance from the center to a vertex along the y-axis, and 'b' is the distance along the x-axis. Understanding this form helps in identifying the ellipse's shape and size.

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Graph Ellipses at Origin

Vertices of an Ellipse

Vertices are the points on the ellipse farthest from the center along the major axis. Given vertices at (0, -6) and (0, 6), the center is at the origin (0,0), and the length of the major axis is 12, so a = 6. Knowing the vertices allows determination of 'a' in the ellipse equation.

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Foci and Vertices of an Ellipse

Using a Point to Find the Ellipse Parameters

Substituting a known point on the ellipse into the standard form equation allows solving for the unknown parameter 'b'. Here, the point (2, 4) lies on the ellipse, so plugging in x=2 and y=4 helps find 'b', completing the equation. This step is essential to fully define the ellipse.

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Finding Equations of Lines Given Two Points

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