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.

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.

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.

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)