Hyperbola: About Hyperbola: Standard Equation of Hyperbola (For CBSE, ICSE, IAS, NET, NRA 2022)

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Title: Hyperbola

  • A spacecraft can use the gravity of a planet to alter its path and propel it at high speed away from the planet and back out into space using a technique called “gravitational slingshot.”
  • If this happens, then the path of the spacecraft is a hyperbola.

About Hyperbola

About Hyperbola
  • It is defined as an open curve having two branches which are mirror images to each other.
  • A hyperbola is two curves that are like infinite bows.
  • Looking at just one of the curves:
  • any point P is closer to F than to G by some constant amount
  • The other curve is a mirror image, and is closer to G than to F.
  • Here, we will be studying about hyperbola and characteristics of such curves.
  • Hyperbolic curves are of special importance in astronomy &space studies.
  • So, the distance from P to F is always less than the distance P to G by some constant amount. (And for the other curve P to G is always less than P to F by that constant amount.)

Standard Equation of Hyperbola

  • The equation of the hyperbola is simplest when the center of the hyperbola is at the origin and the foci are either on the x-axis or on the y-axis.
  • The standard equation of a hyperbola is given as:

Where,

Important Terms and Formulas of Hyperbola

Conic Section

  • The slice must be steeper than that for a parabola, but does not have to be parallel to the cone՚s axis for the hyperbola to be symmetrical.
  • So, the hyperbola is a conic section (a section of a cone) .

Eccentricity (E)

  • Any branch of a hyperbola can also be defined as a curve where the distances of any point from:
  • a fixed point (the focus) , and
  • a fixed straight line (the directrix) is always in the same ratio

Conjugate Hyperbola

2 hyperbolas such that transverse & conjugate axes of one hyperbola are respectively the conjugate & transverse axis of the other are called conjugate hyperbola of each other.

This are conjugate hyperbolas of each other.

Rectangular Hyperbola

Rectangular Hyperbola
  • The rectangular hyperbola is a hyperbola axes (or asymptotes) are perpendicular, or with its eccentricity is .
  • Hyperbola with conjugate axis = transverse axis is a = b example of rectangular hyperbola.

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