Hyperbola

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Session
Hyperbola Session - 1
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Introduction
If S is the focus, ZZ is the directrix and P is
any point on the hyperbola, then by definition
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Illustrative Problem
Find the equation of hyperbola whose focus is (1,
1), directrix is 3x 4y 8 0 and eccentricity
is 2.
Solution
Let S(1, 2) be the focus and P(x, y) be any
point on the hyperbola.
where PM perpendicular distance from P to
directrix 3x 4y 8 0
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Solution Cont.
Ans.
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Equation of The Hyperbola in Standard Form
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Definition of Special Points Lines of the
Equation of Hyperbola
(ii) Transverse and Conjugate Axes
(i) Vertices
(iii) Foci As we have discussed earlier S(ae,
0) and S(ae, 0) are the foci of the hyperbola.
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Definition of Special Points Lines of the
Equation of Hyperbola
(v) Centre The middle point O of AA bisects
every chord of the hyperbola passing through it
and is called the centre of the hyperbola.
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Definition of Special Points Lines of the
Equation of Hyperbola
(vii) Ordinate and Double ordinate
(viii) Latus rectum
A hyperbola is the locus of a point which moves
in such a way that the difference of its
distances from two fixed points (foci) is always
constant.
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Conjugate Hyperbola
The conjugate hyperbola of the hyperbola
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Important Terms
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Auxiliary Circle and Eccentric AngleParametric
Coordinate of Hyperbola
The circle drawn on transverse axis of the
hyperbola as diameter is called an auxiliary
circle of the hyperbola.
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Position of Point with respect to Hyperbola
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Intersection of a Line and a Hyperbola
Point of intersection of line and hyperbola could
be found out by solving the above two equations
simultaneously.
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Intersection of a Line and a Hyperbola
Putting the value of y in the equation of
Hyperbola
This is a quadratic equation in x and therefore
gives two values of x which may be real and
distinct, coincident or imaginary.
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Condition for Tangency and Equation of Tangent in
Slope Form and Point of contact
This is the required condition for tangency.
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Equation of Tangent in Slope Form
Substituting the value of c in the equation y
mx c, we get equation of tangent in slope form.
Equation of tangent
Point of Contact
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Equation of Tangent and Normal in Point Form
Equation of tangent at any point (x1, y1) of
the hyperbola is
Equation of Normal at any point (x1, y1) of
the hyperbola is
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Equation of Tangent and Normal in Parametric
Form
Equation of normal in parametric form is
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Class Exercise - 1
Find the equation to the hyperbola for which
eccentricity is 2, one of the focus is (2, 2) and
corresponding directrix is x y 9 0.
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Solution
Let P(x, y) be any point of hyperbola. Let S(2,
2) be the focus.
This is the required equation of hyperbola.
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Class Exercise - 2
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Solution
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Solution contd..
Shifting the origin at (1, 2) withoutrotating
the coordinate axes, i.e.
Put x 1 X and y 2 Y
Centre The coordinates of centre with respect to
new axes are X 0 and Y 0.
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Solution contd..
Length of axes
Length of transverse axes 2b
Length of conjugate axes 2a
Eccentricity
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Solution contd..
Length of latus rectum
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Solution contd..
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Class Exercise - 3

• Find the equation of hyperbola whose
• direction of axes are parallel to
• coordinate axes if
• vertices are (8, 1) and (16, 1) and focus is
  (17, 1) and
• focus is at (5, 12), vertex at (4, 2) and centre
  at (3, 2).

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Solution
Let x 4 X, y 1 Y.
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Solution contd..
As per definition of hyperbola
a Distance between centre and vertices
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Abscissae of focus in new coordinates system isX
ae, i.e. x 4 12e
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Solution contd..
Let x 3 X, y 2 Y.
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Solution contd..
i.e. x 3 e (As a 1)
x e 3
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Class Exercise - 4
Find the equations of the tangents to the
hyperbola 4x2 9y2 36 which are parallel to
the line 5x 3y 2.
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Solution
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Class Exercise - 5
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Solution
Let the tangent (i) intersect the x-axis at A and
y-axis at B respectively. Let P(h, k) be the
middle point of AB.
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Solution contd..
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Class Exercise - 6
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Solution
The equation of the given line is lx my n
0 ...(ii)
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Solution contd..
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Class Exercise - 7
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Solution
Hence, answer is (c).
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Class Exercise - 8
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Solution
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Solution contd..
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Class Exercise - 9
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Solution
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Solution contd..
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Class Exercise - 10
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Solution
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Solution contd..
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