Question

Given vertices \((\pm a, 0)\) and eccentricity \(e,\) what are the coordinates of the foci of an ellipse and a hyperbola?

Short answer

Answer: The foci of both the ellipse and the hyperbola are located at \((-ea, 0)\) and \((ea, 0)\).

Step-by-step solution

1. Identify the given information:

We know the vertices of the ellipse and hyperbola are at \((\pm a, 0)\). The eccentricity is given as \(e\).

2. Find the foci of the ellipse:

For the ellipse, we have the equation \(c^2 = a^2 - b^2\). We can relate the eccentricity, vertices, and foci by the equation \(e = \frac{c}{a}\). Therefore, we need to solve for \(c\): $$c = ea$$ Now, we will substitute this into the equation for the ellipse: $$(ea)^2 = a^2 - b^2$$ The coordinates of the foci of the ellipse are \((-c, 0)\) and \((c, 0)\). Therefore, we need to substitute the value of \(c\) that we found: $$\left(-ea, 0\right),\left(ea, 0\right)$$

3. Find the foci of the hyperbola:

For the hyperbola, we have the equation \(c^2 = a^2 + b^2\). Using the relation we found for the eccentricity \(e = \frac{c}{a}\), we can solve for \(c\): $$c = ea$$ We substitute this into the equation for the hyperbola:: $$(ea)^2 = a^2 + b^2$$ The coordinates of the foci of the hyperbola are also \((-c, 0)\) and \((c, 0)\). Thus, the foci of the hyperbola will have the same coordinates as the ellipse: $$\left(-ea, 0\right),\left(ea, 0\right)$$

Key concepts

Eccentricity

Eccentricity is a key concept that defines the shape and characteristics of conic sections. It's a non-negative real number denoted by the symbol \(e\), which tells us how 'stretched out' a conic section is. For an ellipse, the eccentricity has to be between 0 and 1, where an \(e = 0\) represents a perfect circle. The closer the value is to 1, the more elongated the ellipse.

On the other hand, for a hyperbola, the eccentricity is always greater than 1. This is because a hyperbola is essentially an 'open' curve that extends to infinity, unlike the closed curve of an ellipse. The higher the value of \(e\), the narrower and more 'open' the hyperbola appears.

In the context of the given exercise, the eccentricity \(e\) is used to determine the distance between the center of the conic section and its foci. This shows that eccentricity isn't just a measure of shape but also directly impacts the geometry of the section.

Vertices of Conic Sections

Vertices are significant points on conic sections that provide a reference for understanding their shapes and dimensions. For ellipses, the vertices are where the curve is farthest from the center in the major axis direction. An ellipse has two pairs of vertices, with each pair lying on one of its two axes. The major vertices are on the longer axis (major axis), while the minor vertices are on the shorter one (minor axis).

For hyperbolas, the vertices are the points closest to the center on each branch. They are located where the hyperbola intersects its transverse axis. In both cases, \( \pm a \) represent the distances from the center to the vertices along the major or transverse axis.

In the exercise, knowing the vertices helps us understand the size of the ellipse or hyperbola and is pivotal in determining the location of other critical points, like the foci. The vertices, along with the center, let us visualize the conic section's extent and orientation in the coordinate plane.

Conic Sections Equations

The equations for conic sections are essential for describing their geometry analytically. These equations allow us to identify the set of points that make up the conic sections and determine their properties.

For an ellipse, the standard form of the equation is \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1,\] where \(a\) is the semi-major axis and \(b\) is the semi-minor axis. For a hyperbola, the equation takes a similar form but with a key difference: \[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1,\] or \[\frac{y^2}{b^2} - \frac{x^2}{a^2} = 1,\] depending on the orientation of the hyperbola.

The distance between the center and the foci is denoted by \(c\), and connecting it with the vertices, we form the equations \(c^2 = a^2 - b^2\) for the ellipse and \(c^2 = a^2 + b^2\) for the hyperbola. Understanding these equations is crucial for solving problems involving conic sections, such as finding the coordinates of the foci, as demonstrated in the exercise. They encapsulate the relationship between the key components of each conic section, including the vertices, foci, and eccentricity.