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Hyperbola in Conic Sections




Hyperbola

  • A hyperbola is the set of all points in a plane, the difference of whose distances from two fixed points in the plane is a constant.
  • Foci: The two points are called the Foci
  • Centre of Hyperbola: The midpoint of the line segment joining the foci is called the centre of the hyperbola
  • Transverse axis: The line through the foci is called Major axis
  • Conjugate axis: The line segment through the center and perpendicular to the Transverse axis
  • Vertices: The point at which hyperbola hits the transverse axis

  • We denote the distance between the two foci by 2c, the distance between two vertices (the length of the transverse axis) by 2a
  • we define the quantity b as $b=\sqrt {c^2 -a^2}$

Eccentricity of Hyperbola

It is the ratio of the distances from the centre of the Hyperbola to one of the foci and to one of the vertices of the Hyperbola
It is denoted by letter e
$e= \frac {c}{a} =\frac {1}{a} \sqrt {a^2 + b^2} = \sqrt {1 + \frac {b^2}{a^2}}$
So c=ae
For an Hyperbola, e > 1.
Also we can write c=ae

Standard equation of Hyperbola when Foci lies on x-axis and center at origin


$\frac {x^2}{a^2} - \frac {y^2}{b^2} =1$
Here a> b
Foci can be found as (c,0) and (-c,0). It is on x-axis
$c^2 =a^2 + b^2$
Length of Transverse axis =2a
Vertices (a,0) and (-a,0)
Length of Conjugate axis =2b

Standard equation of Hyperbola when Foci lies on y-axis and center at origin


$\frac {y^2}{a^2} - \frac {x^2}{b^2} =1$
here a> b
Foci can be found as (0, c) and (0, -c). It is on y-axis
$c^2 =a^2 +b^2$
Length of Transverse axis =2a
Vertices (0, a) and (0, -a)
Length of Conjugate axis =2b

Important Observtions

(i)Hyperbola is symmetric with respect to both the axes, since if (x, y) is a point on the hyperbola, then (– x, y), (x, – y) and (– x, – y) are also points on the hyperbola
(ii) The foci are always on the transverse axis. It is the positive term whose denominator gives the transverse axis

Latus Rectum of Hyperbola

Latus rectum of hyperbola is a line segment perpendicular to the transverse axis through any of the foci and whose end points lie on the hyperbola
$l= \frac {2b^2}{a}=2a(1-e^2)$

Parametric Form of Hyperbola

$x = a sec \theta$ and $y = b tan \theta$ together represent the Hyperbola.

Position of a Point with Respect to a Hyperbola

We assume the middle portion to be inside
A point (x1,y1) lies inside the Hyperbola S ≡$\frac {x^2}{a^2} - \frac {y^2}{b^2} -1=0$ if $S_1 ≡\frac {x_1^2}{a^2} - \frac {y_1^2}{b^2} -1$ is negative (S1 < 0).
If S1 > 0, the point lies outside the Hyperbola.
If S1 = 0, the point lies on the Hyperbola.

Equation of Tangent in Hyperbola

Equation of tangent at $P(x_1,y_1)$ in a Hyperbola
$\frac {x^2}{a^2} - \frac {y^2}{b^2} =1$
Equation of tangent
$\frac {xx_1}{a^2} - \frac { yy_1}{b^2} =1$
Another form
The equation y = mx + c is a tangent to the ellipse if $c^2 = a^2m^2- b^2$

Equation of Normal of Hyperbola

Normal to any point on the Hyperbola is the line which passes through point and its perpendicular to the tangent at that point
Equation of Normal
$\frac {x -x_1}{x_1/a^2} =\frac {y -y_1}{y_1/b^2} $

Pair of Tangents


Two tangents, PQ and PR, can be drawn from a point $P(x_1, y_1)$ to the Hyperbola $S = \frac {y^2}{a^2} - \frac {x^2}{b^2} -1 = 0$.
The combined equation is
$(x^2/a^2 -y^2/b^2 -1)(x_1^2/a^2 - y_1^2/b^2 -1)= (xx_1/a^2 - yy_1/b^2 -1 )^2$
This can be written like below to remember easily
$SS_1 = T^2$,
Where S = 0 is the equation of the circle, T = 0 is the equation of tangent at $(x_1, y_1)$, and S1 is obtained by replacing x by x1 and y by y1 in S.

Chord of Contact

The chord joining the two points of contact of tangents to a circle drawn from any point P is called the chord of contact of P with respect to the given circle.
Equation of Chord of Contact
$\frac {xx_1}{a^2} - {yy_1}{b^2} =1$

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