Problem 2 - Algebra 2

Question

A hyperbola has foci at \((4,0)\) and \((-4,0) .\) The value of \(a\) is \(1 .\) Write an equation for the hyperbola.

Step-by-Step Solution

Answer

The equation of the hyperbola is \( x^2 - \frac{y^2}{15} = 1 \).

Step 1: Identify the Center

The center of the hyperbola, given the foci at (4, 0) and (-4, 0), is the midpoint of these foci. The center is at the origin, (0, 0).

Step 2: Determine the Distance Between Foci

Calculate the distance between the foci, which is the distance from (4, 0) to (-4, 0). This distance is 8. Thus, we find that 2c = 8, so c = 4.

Step 3: Use the Relationship Among a, b, and c

Recall the hyperbola relationship: \[ c^2 = a^2 + b^2 \] Substitute the known values (a=1, c=4): \[ 4^2 = 1^2 + b^2 \]\[ 16 = 1 + b^2 \]\[ b^2 = 15 \]

Step 4: Formulate the Standard Equation of the Hyperbola

For a horizontal hyperbola centered at (0, 0), with a=1 and b calculated: \[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \] Substituting the value of a and b gives: \[ \frac{x^2}{1} - \frac{y^2}{15} = 1 \]

Step 5: Write the Final Equation

The standard equation is simplified to: \[ x^2 - \frac{y^2}{15} = 1 \]

Key Concepts

Equation of Hyperbola

To understand the equation of a hyperbola, we first need to recognize the standard form. A hyperbola in its simplest form is expressed as \[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \]when centered at the origin and opening horizontally. Here, \(x^2\) and \(y^2\) indicate the axes of symmetry, where \(a\) and \(b\) are distances that define the hyperbola’s size and shape. The value \(a\) relates to the distance from the center to the vertices, while \(b\) impacts the curvature and distance of the hyperbola's arms. Both parameters ultimately determine the equation's structure by extending or compressing the curves. For vertical hyperbolas, the equation would be:\[ \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \] In our example, we have \(x^2 - \frac{y^2}{15} = 1\), indicating a horizontal hyperbola because the \(x\) term comes first and is positive. The structure of this equation is crucial for graphing and understanding the hyperbola considerably.

Conic Sections

Conic sections involve slicing a cone in different ways to form unique shapes: circles, ellipses, parabolas, and hyperbolas. Imagine slicing a cone vertically at an angle; the intersecting shape impacts the geometry. - **Circle**: A cross-section parallel to the cone's base forms a circle, noted by an equation like \(x^2 + y^2 = r^2\), where \(r\) is the radius.- **Ellipse**: Slicing at an angle thinner than the base forms an ellipse. It can be expressed as \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\).- **Parabola**: Cutting parallel to the cone’s side produces a parabola, described in simple terms as \(y = ax^2 + bx + c\).- **Hyperbola**: Slicing perpendicular produces two curves facing outward, known as a hyperbola. Its equation takes form based on its orientation, either horizontally or vertically.Hyperbolas are among the most interesting conics due to their open, divergent shape. They embody unique symmetry and are applied in modeling various physical phenomena like satellite dishes and natural sound wave patterns.

Foci of Hyperbola

Foci are key points within hyperbolas that define their shape and orientation. A hyperbola always has two foci, located along its major axis. These points lie outside the hyperbola, at a distance \(c\) from the center. For our problem, the foci are at (4, 0) and (-4, 0), indicating horizontal orientation.The mathematical relationship connecting foci and hyperbolas is given by:\[ c^2 = a^2 + b^2 \]In our example, since \(2c = 8\), we deduce \(c = 4\). Even though the foci sit beyond the vertices, their role is integral. Observing how the difference in distances from points on the hyperbola to each focus remains constant helps us keep track of the structure and alignment. These characteristics illuminate insights into the hyperbola's openness and the distance between its arms. Understanding these aspects helps apply hyperbolas in tracking planetary orbits, engineering optics, and more.