Mathematics. Session. Hyperbola Session - 1. Introduction. If S is the focus, ZZ´ is the directrix and P is any point on the hyperbola, then by definition. Question. Illustrative Problem.
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Session - 1
If S is the focus, ZZ´ is the directrix and P is any point on the hyperbola, then by definition
Find the equation of hyperbola whose focus is (1, 1), directrix is 3x + 4y + 8 = 0 and eccentricity is 2.
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
(ii) Transverse and Conjugate Axes
(iii) Foci : As we have discussed earlier S(ae, 0) and S´(–ae, 0) are the foci of the hyperbola.
are two directrices of the hyperbola and
their equations are
For the hyperbola we haveDefinition 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.
SP = ex – a
S´P = ex + aDefinition 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.”
Of hyperbola, then its auxiliary circle is x2 + y2 = a2
= eccentric angle
are known as parametric equation
of hyperbola.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.
[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.
and given line is y = mx + cCondition for Tangency and Equation of Tangent in Slope Form and Point of contact
This is the required condition for tangency.
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
Equation of tangent at any point
(x1, y1)of the hyperbola is
Equation of Normal at any point
(x1, y1)of the hyperbola is
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.
Find the coordinates of centre, lengths of the axes, eccentricity, length of latus rectum, coordinates of foci, vertices and equation of directrices of the hyperbolaClass Exercise - 2
Coordinates of foci with respect to old axes are(1, 5) and (1, –1).
Vertices: The coordinates of vertices with respect tonew axes are X = 0 and , i.e. X = 0 andSolution contd..
Length of latus rectum
The coordinates of axes with respect toold axes are x – 1 = 0, i.e. x = 1 and
Directrices: The equation of directrices with respectto new axes are , i.e. .
The equation of directrices with respect toold axesare , i.e. y = 3 and y = 1.Solution contd..
As per definition of hyperbola
a = Distance between centre and vertices
Abscissae of focus in new coordinates system isX = ae, i.e. x – 4 = 12e
Find the equations of the tangents to the hyperbola 4x2 – 9y2 = 36 which are parallel to the line 5x – 3y = 2.
Find the condition that the linelx + my + n = 0 will be normal to the hyperbolaClass Exercise - 6
The equation of the given line is
lx + my + n = 0 ...(ii)
Let where be two points on thehyperbola If (h, k) is the point of intersection of the normals at P and Q, then k is equal to
(c) (d)Class Exercise - 9