Understanding the Center, Vertices, Foci, and Co-vertices of a Hyperbola
A hyperbola is a conic section formed by the intersection of a plane with the surface of a double cone. It has several key features including the center, vertices, foci, and co-vertices. Let's explore one such hyperbola and understand its geometric significance using the equation 4x2-8y2-12x 16y-160.
Transforming the Equation
The given equation 4x2-8y2-12x 16y-160 can be rewritten in a more recognizable form to identify the center, vertices, foci, and co-vertices. Let's start by completing the square for both x and y terms.
Steps to Transform the Equation
Group terms involving x and y:(4x2-12x) - (8y2-16y) 16.Complete the square for x terms:
4(x2-3x) - 8y2 16y 16.Shift the constant to the right and factor out the 4 from x terms.
4(x2-3x (3/2)2- (3/2)22 16y 16.Complete the square for y terms:
4(x 2-3x (3/2)2 - (3/2)2) - 8(y2-2y 1 - 1) 16.Adjust the equation:
4(x2-3x (3/2)2) - 4(3/2)2 - 8(y2-2y 1) 8 16.Simplify the equation and adjust the constants.
4(x - 3/2)2 - 9 - 8(y - 1)2 8 16.Simplify further:
4(x - 3/2)2 - 8(y - 1)2 16 9 - 8.Final equation:
4(x - 3/2)2 - 8(y - 1)2 17.Divide by 17 to normalize the equation.
(x - 3/2)2/ (17/4) - (y - 1)2/ (17/8) 1.
Identifying Key Features of the Hyperbola
Now, the equation is in the standard form of a hyperbola with center (h, k), which can be written as:
(x - h)2/a2 - (y - k)2/b2 1.
From the given equation:
Center (h, k) (3/2, 1) a2 17/4 rarr; a radic;(17/4) radic;17/2 b2 17/8 rarr; b radic;(17/8) radic;17/ radic;2 radic;17/1.414 ≈ 1.374Vertices, Foci, and Co-vertices
The vertices, foci, and co-vertices of a hyperbola can be determined using the following formulas:
Vertices: ((h±a, k)) Foci: ((h±radic;(a2 b2), k)) Co-vertices: ((h, k±b))Calculations for the Given Hyperbola
Vertices: V1 (3/2 radic;17/2, 1) ≈ (3.68, 1) V2 (3/2 - radic;17/2, 1) ≈ (-0.18, 1) Foci: F1 (3/2 radic;((17/4) (17/8)), 1) ≈ (3/2 radic;23.5/2, 1) ≈ (3.38, 1) F2 (3/2 - radic;23.5/2, 1) ≈ (-0.88, 1) Co-vertices: C1 (3/2, 1 radic;17/1.414) ≈ (1.5, 2.374) C2 (3/2, 1 - radic;17/1.414) ≈ (1.5, -0.374)Visualizing the Hyperbola
A hyperbola can be visualized as two separate parabolas that never intersect. Using graphing software, we can plot the hyperbola and its key features:
The plot shows the hyperbola centered at (1.5, 1) with vertices around (3.68, 1) and (-0.18, 1), foci around (3.38, 1) and (-0.88, 1), and co-vertices around (1.5, 2.374) and (1.5, -0.374).
Conclusion
Understanding the center, vertices, foci, and co-vertices of a hyperbola is fundamental in conic sections and can be crucial in various fields such as physics, engineering, and mathematics. By transforming the given equation and applying the formulas for these key features, we can gain insight into the structure and properties of the hyperbola.
For further exploration, consider visiting resources on hyperbolas and conic sections:
Wikipedia - Hyperbola MathWorld - Hyperbola MathIsFun - Hyperbola