A U-Shaped curve who any point is at an equal distance from a fixed point, also known as a focus, and forms a straight line, also known as directrix, is termed as the parabola.
Parabola is defined as the locus of a point that moves in a way such that distance from a fixed point is equivalent to the distance from a fixed line (directrix). This topic can be explored more on Cuemath.
Here,
The fixed point is termed as focus, whereas
The fixed-line is termed the directrix.
What is Know As the Axis of Symmetry of Parabola?
A vertical line that splits the conic section parabola into two identical fractions is termed its axis of symmetry. The axis of symmetry of the parabola always passes through its vertex.
What is Known As the Vertex of Parabola?
A point where the parabola crosses its axis of symmetry is termed its vertex.
The vertex of the parabola will be formed at the bottom of its ‘U’ shaped curve if there is a positive coefficient of the x² term.
Contrarily, the vertex will be formed at the top of its ‘U’ shaped curve if there is a negative coefficient of the x² term.
Real-Life Applications of Parabola
- The stretched arc of a rocket launch is the most common example of a parabola.
- Water in The Bellagio’s fountains in Las Vegas is in the shape of a parabola.
- Parabola is widely seen in architectural and engineering projects.
- The main suspension cables of the Golden Gate Bridge in San Francisco, California is in the shape of a parabola.
- The parabolic trajectories are most widely seen from decades.
- The architectural structure named “The Parabola in London in 1962” has a copper roof with parabolic and hyperbolic lines.
Equation of a Hyperbola
A two-branched open curve obtained by the intersection of a circular cone and plane that crosses both the nappes of a cone is termed a hyperbola.
As a plane, a hyperbola is defined as the path (locus of a point) moving in such a way that the ratio of a distance from the fixed point (the focus) to the distance from a fixed line (the directrix) is a constant greater than 1.
Hyperbola, due to its symmetry has two foci. Now we will learn about equation of hyperbola.
What is the Equation of a Hyperbola?
A hyperbola drawn on x-y graph (centred over the x-axis and y – axis), its standard equation when opening left and right is given as:
(x – h)²a² – (y – k)²b² = 1
A hyperbola drawn on x-y graph (centred over the x-axis and y – axis), its standard equation when opening upward and downward is given as:
(y – k)²b² – (x – h)²a² = 1
Example: Calculate the vertex of a following hyperbola:
x²9 – y²16 = 1
Solution: Generally, the vertex of a hyperbola x²a² – y²b² = 1 is given at (-a, 0) and (a,0).
Here,
a² = 9
a = 9 = 3
Hence, the vertex of a given hyperbola is (-3,0) and (3,0)
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