What is a Hyperbola?
Formula:
The formula is like so: x^2/a^2 - y^2/a^2 = 1
There are two main types of hyperbola: East-West Hyperbola (formula above) and North-South Hyperbola, or horizontal and vertical hyperbola. (swap the x^2 and y^2).
The formula is like so: x^2/a^2 - y^2/a^2 = 1
There are two main types of hyperbola: East-West Hyperbola (formula above) and North-South Hyperbola, or horizontal and vertical hyperbola. (swap the x^2 and y^2).
Origin
Discoverer:
The hyperbola was probably discovered by Menaechmus, an ancient Greek mathematician and geometer. He was famous for his discovery of the conic sections and his solution to the problem of doubling the cube using the parabola and the hyperbola. Euler’s law follows the Hyperbola format.
The hyperbola was probably discovered by Menaechmus, an ancient Greek mathematician and geometer. He was famous for his discovery of the conic sections and his solution to the problem of doubling the cube using the parabola and the hyperbola. Euler’s law follows the Hyperbola format.
Properties
Properties you may notice:
The two halves of the hyperbola can never meet, and can only intersect with either the Y or X axis, never both. The hyperbola also follows trigonometry since it uses the euler’s formula and x and y can be replaced with cosh and sinh. Also if p is on any point of the curve automatically pr and ps are drawn parallel to the asymptotes. Asymptotes lie on the same points of the intersection of the foci. The line joining the vertices is also known as the transverse axis and its perpendicular line is called the conjugate axis. If this is appearing in the H.S.C you know these as if you forget these YOU WILL DIE!!!!!!!!!!!!. And be shunned by society for the rest of your life.
The two halves of the hyperbola can never meet, and can only intersect with either the Y or X axis, never both. The hyperbola also follows trigonometry since it uses the euler’s formula and x and y can be replaced with cosh and sinh. Also if p is on any point of the curve automatically pr and ps are drawn parallel to the asymptotes. Asymptotes lie on the same points of the intersection of the foci. The line joining the vertices is also known as the transverse axis and its perpendicular line is called the conjugate axis. If this is appearing in the H.S.C you know these as if you forget these YOU WILL DIE!!!!!!!!!!!!. And be shunned by society for the rest of your life.
Construction
Sketching:
Graphing on an excel spreadsheet:
- Say an equation was given, e.g. x^2/9 - y^2/16 = 1
- From this equation, we know that a = sqrt(9) = 3, and b = sqrt(16) = 4.
- To find the eccentricity (the curvature) of the hyperbola, we need to find c, from this equation: c^2 = a^2 + b^2 (Yes, it’s the same formula as the Pythagorean theorem) - c^2 = 9 + 16, c^2 = 25, c = 5.
- The eccentricity is c/a, so it is 5/3 in this example.
- Because x^2=(x-0)^2 and y^2=(y-0)^2, the centre of the graph is (0,0).
- The vertices (a and -a) are at co-ordinates (0,a) and (0,-a), so they would be (0,3) and (0,-3).
- The foci (c and -c) are at co-ordinates (0,c) and (0,-c), so they would be (0,5) and (0,-5).
Graphing on an excel spreadsheet:
Real-life Applications
Aircraft:
If an airplane is travelling on the speed of sound, the sonic boom formed will be in the shape of a cone behind the plane. If the plane is parallel to the ground, eventually the cone will intersect with the ground creating a hyperbola.
Sundial:
On any given day, the sun revolves in a circle, and its rays occasionally strike the point tracing out a cone of light. To put it in basic terms, the shadow of the tip of the pole in a sundial traces out a hyperbola on the ground over the course of the day. The shape of this hyperbola varies with the geographical latitude and with the time of the year, since those factors affect the cone of the sun's rays relative to the horizon.
Microscopes:
Hyperbola's play a large role in scopes (microscopes and telescopes) as where the hyperbola is used to bend light into a focal point. And when your looking through it you are placing your eye in a well planned focal point that allows the light from an unseen object as its light is focused into a single point.
Space:
The orbit of sattelites and planets are calculated with hyperbola's as hyperbola is based on gravity.
If an airplane is travelling on the speed of sound, the sonic boom formed will be in the shape of a cone behind the plane. If the plane is parallel to the ground, eventually the cone will intersect with the ground creating a hyperbola.
Sundial:
On any given day, the sun revolves in a circle, and its rays occasionally strike the point tracing out a cone of light. To put it in basic terms, the shadow of the tip of the pole in a sundial traces out a hyperbola on the ground over the course of the day. The shape of this hyperbola varies with the geographical latitude and with the time of the year, since those factors affect the cone of the sun's rays relative to the horizon.
Microscopes:
Hyperbola's play a large role in scopes (microscopes and telescopes) as where the hyperbola is used to bend light into a focal point. And when your looking through it you are placing your eye in a well planned focal point that allows the light from an unseen object as its light is focused into a single point.
Space:
The orbit of sattelites and planets are calculated with hyperbola's as hyperbola is based on gravity.