Problem 26 - Algebra 2

Question

Write the equation of a hyperbola with the given foci and vertices. foci \(( \pm \sqrt{5}, 0),\) vertices \(( \pm 2,0)\)

Step-by-Step Solution

Answer

The equation of the hyperbola with foci at \(( \pm \sqrt{5}, 0)\) and vertices at \(( \pm 2,0)\) is \( \frac{x^2}{4} - y^2 = 1 \).

Step 1: Identify coordinates of vertices and foci

The vertices are at \(( \pm 2,0)\) and the foci are at \( ( \pm \sqrt{5}, 0) \). Therefore, the lengths from the center to each vertex (a) and from the center to each focus (c) are 2 and \( \sqrt{5} \) respectively.

Step 2: Identify the values of a and c

The center of the hyperbola is at the origin (0,0). The distance from the center to a vertex is 'a' and the distance from the center to a focus is 'c'. Given the coordinates, we can see that a = 2 and c = \( \sqrt{5} \) .

Step 3: Determine the value of b using the relationship \( c^2 = a^2+b^2 \)

Using the relationship \( c^2 = a^2+b^2 \) , which holds for hyperbolas, we can find the value of b. Substituting the given values of a and c in this formula we get \( (\sqrt{5})^2 = 2^2 + b^2 \). This simplifies to 5 = 4 + \( b^2 \), and by rearranging the equation we find \( b^2 = 1\).

Step 4: Write the equation of the hyperbola

The standard form of the equation of a hyperbola opening left and right is \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \). Substituting the values we've found gives the equation \( \frac{x^2}{2^2} - \frac{y^2}{1^2} = 1 \) .

Key Concepts

Hyperbola

A hyperbola is a type of conic section that you obtain by slicing a cone with a plane. It consists of two separate curves known as the branches of the hyperbola. Unlike circles or ellipses, hyperbolas have a unique pair of distinct lines as their asymptotes. These lines are imaginary boundaries that the hyperbola gets closer to but never meets. In graphical terms, hyperbolas look like an open pair of mirrored arches that face away from one another. This shape arises when the slice is made at an angle such that the plane intersects both sides of the cone.

Vertices

The vertices of a hyperbola are key points that help determine its shape. They lie on the x-axis for a horizontally oriented hyperbola and on the y-axis for a vertically oriented one. The vertices indicate the points at which the branches are closest to each other. For a hyperbola with its center at the origin, the coordinates of the vertices are often given as

• (±a, 0) for a horizontal hyperbola
• (0, ±a) for a vertical hyperbola

where 'a' is the distance from the center to each vertex. This distance is crucial in understanding the scale and spacing of the hyperbola.

Foci

The foci (plural of focus) of a hyperbola are special points used to define and construct the shape of the curve. They are important for determining the properties of the hyperbola. Unlike the vertices, foci are located outside the curve along the axis of symmetry. For a hyperbola centered at the origin, the

• horizontal hyperbola has foci at (±c, 0)
• vertical hyperbola has foci at (0, ±c)

where 'c' represents the distance from the center to a focus. This relationship helps establish the relative position of the branches and emphasizes the stretching of the hyperbola.

Equation of Hyperbola

Writing the equation of a hyperbola involves using the standard form depending on its orientation. For horizontally oriented hyperbolas, the standard form is \[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \] Conversely, if the hyperbola is vertical, the equation becomes \[ \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \] In these formulas:

• 'a' is the distance to the vertices from the center
• 'b' is related to 'a' and 'c' by the equation \( c^2 = a^2 + b^2 \)
• 'c' is the distance from the center to a focus

These forms highlight how the variables relate to each other and provide all the details needed to draw the hyperbola or understand its shape. They enable you to visualize the hyperbola's orientation and scale based on its given characteristics.