53–57. Conic sections
b. Use analytical methods to determine the location of the foci, vertices, and directrices.
4x² + 8y² = 16

Verified step by step guidance

Rewrite the given equation \(4x^{2} + 8y^{2} = 16\) in standard form by dividing both sides by 16 to get \(\frac{4x^{2}}{16} + \frac{8y^{2}}{16} = 1\), which simplifies to \(\frac{x^{2}}{4} + \frac{y^{2}}{2} = 1\).

Identify the type of conic section. Since both \(x^{2}\) and \(y^{2}\) terms are positive and added, this is an ellipse with the standard form \(\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1\) where \(a^{2} = 4\) and \(b^{2} = 2\).

Determine the vertices. For an ellipse centered at the origin with the \(x\)-axis as the major axis (since \(a^{2} > b^{2}\)), the vertices are located at \((\pm a, 0)\), so here they are at \((\pm 2, 0)\).

Calculate the focal distance \(c\) using the relationship \(c^{2} = a^{2} - b^{2}\). Substitute the values to find \(c^{2} = 4 - 2\) and then find \(c\) by taking the square root.

Find the foci at \((\pm c, 0)\) along the major axis. The directrices are vertical lines given by \(x = \pm \frac{a^{2}}{c}\). Write these equations to complete the location of the directrices.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

Key Concepts

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

Standard Form of Conic Sections

Conic sections like ellipses, hyperbolas, and parabolas can be expressed in standard forms that reveal their geometric properties. For the given equation, rewriting it in standard form helps identify the type of conic and its key parameters such as center, axes lengths, and orientation.

Foci, Vertices, and Directrices of an Ellipse

In an ellipse, the foci are two fixed points inside the curve whose sum of distances to any point on the ellipse is constant. Vertices are points where the ellipse intersects its major axis. Directrices are lines related to the ellipse’s eccentricity, used to define the curve analytically.

Analytical Methods for Locating Key Features

Analytical methods involve algebraic manipulation and formulas to find the foci, vertices, and directrices from the conic’s equation. This includes completing the square, identifying parameters like a, b, and c, and applying relationships such as c² = a² - b² for ellipses.

Recommended video: