Problem 55 - Algebra 2

Question

Write an equation of an ellipse in standard form with center at the origin and with the given characteristics. \(c^{2}=68,\) vertex \((0,-18)\)

Step-by-Step Solution

Answer

The equation of the ellipse is \(\frac{x^{2}}{392} + \frac{y^{2}}{{324}} = 1\)

Step 1: Determine value of \(a\) or \(b\)

Since this ellipse has a vertex at (0,-18), the value for one the semiaxis is 18, that is, either \(a = 18\) or \(b = 18\).

Step 2: Find other semi-axis

We know that \(c^2 = 68\) and \(c^{2} = |a^{2} - b^{2}|\). Since \(c\) has to be less than \(a\) and \(b\) for an ellipse, we can assume that \(b = 18\) and that \(|a^{2} - b^{2}| = c^{2} = 68\). Solving this gives us \(a^{2} = 68 + 18^{2} = 68 + 324 = 392\), so \(a = \sqrt{392}\).

Step 3: Write the equation in standard form

Now, combining the values of \(a\) and \(b\) into the standard form for the equation of an ellipse we have \(\frac{x^{2}}{(\sqrt{392})^{2}} + \frac{y^{2}}{{18}^{2}} = 1\), which simplifies to \(\frac{x^{2}}{392} + \frac{y^{2}}{{324}} = 1\).

Key Concepts

Ellipse Characteristics

An ellipse is a fascinating shape found in mathematics, and its characteristics are essential in understanding its geometry. An ellipse is defined by two main parameters: the semi-major axis and the semi-minor axis. These correspond to the longest and shortest radii of the ellipse, respectively. In the exercise, we have a vertex located at \(0, -18\), indicating that it serves as one end of a semi-axis, which in this context, is the vertical semi-axis with length 18.

• Semi-major and Semi-minor Axes: In our ellipse, since we found \(b = 18\), this means the semi-minor axis is vertical.
• Foci: The foci are two fixed points inside the ellipse. The exercise provides \(c^2 = 68\), with \(|a^2 - b^2| = c^2\).

Knowing the lengths of these axes is crucial because they help us construct the standard form equation of the ellipse.

Conic Sections

Ellipses belong to an interesting family of shapes known as conic sections, which also include parabolas, hyperbolas, and circles. These shapes emerge from slicing a double-napped cone at various angles and orientations.

• Origins of Conic Sections: An ellipse forms when a plane intersects a cone at an angle less steep than its side but does not pass through the base.
• Properties: All conic sections follow properties defined by their respective equations, dictating their curvature and orientation in a plane.

Understanding ellipses as part of conic sections makes it easier to see how they relate to parabolas and hyperbolas, enriching our comprehension of coordinate geometry. This broad classification underlines their significance in not just geometry but also in physics and astronomy.

Coordinate Geometry

Coordinate geometry, or analytic geometry, is the study of geometry using a coordinate system. It allows mathematical representation of geometric figures using algebraic equations.

• Standard Equation of an Ellipse: The equation \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) is used to describe ellipses centered at the origin on the Cartesian plane.
• Derivation in Exercise: In the exercise, the standard form equation derived was \ \frac{x^2}{392} + \frac{y^2}{324} = 1 \, showcasing the calculated axes lengths.
• Applications: These equations are crucial in various fields, such as physics for orbits, engineering for stress analysis, and computer graphics for rendering curves.

Through the lens of coordinate geometry, we can sense the elegance in how algebra meets geometry to provide a clear understanding and visualization of shapes like ellipses interacting on a plane.