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Ch.12 - Parametric and Polar Curves
Briggs - Calculus: Early Transcendentals 3rd Edition
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Tutti i libri di testoBriggs 3rd EditionCh.12 - Parametric and Polar CurvesProblema 12.4.43
Capitolo 12, Problema 12.4.43

39–50. Equations of ellipses and hyperbolas Find an equation of the following ellipses and hyperbolas, assuming the center is at the origin. 
An ellipse with vertices (±5, 0), passing through the point (4, 3/5)

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Identify the standard form of the ellipse equation centered at the origin with a horizontal major axis: \(\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1\).
From the vertices given as \((\pm 5, 0)\), determine the value of \(a\) since the vertices lie along the major axis. Here, \(a = 5\), so \(a^{2} = 25\).
Substitute \(a^{2} = 25\) into the ellipse equation to get \(\frac{x^{2}}{25} + \frac{y^{2}}{b^{2}} = 1\).
Use the point \((4, \frac{3}{5})\) that lies on the ellipse to substitute \(x = 4\) and \(y = \frac{3}{5}\) into the equation, resulting in \(\frac{4^{2}}{25} + \frac{(\frac{3}{5})^{2}}{b^{2}} = 1\).
Solve the resulting equation for \(b^{2}\) to find the value of \(b^{2}\), which completes the equation of the ellipse.

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Standard Form of an Ellipse Equation

An ellipse centered at the origin with a horizontal major axis has the equation (x²/a²) + (y²/b²) = 1, where 'a' is the distance from the center to each vertex along the x-axis, and 'b' is the distance along the y-axis. Knowing the vertices helps determine 'a', which is essential for writing the ellipse equation.
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Parameterizing Equations of Circles & Ellipses

Using a Point on the Ellipse to Find Parameters

Substituting a known point on the ellipse into the standard equation allows solving for the unknown parameter 'b'. This step ensures the ellipse passes through the given point, refining the equation to fit the specific ellipse described.
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Finding Volume Using Disks

Relationship Between 'a', 'b', and the Shape of the Ellipse

The values of 'a' and 'b' determine the ellipse's shape and size. 'a' is the semi-major axis length, and 'b' is the semi-minor axis length. Understanding their relationship helps in interpreting the ellipse's geometry and verifying the correctness of the equation.
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Finding Area Between Curves on a Given Interval
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