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 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

Solution Cont. Ans.

Equation of The Hyperbola in Standard Form

Definition of Special Points Lines of the Equation of Hyperbola (i) Vertices (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.

Definition of Special Points Lines of the Equation of Hyperbola (iv) Directrices :The lines zk and z´k´ are two directrices of the hyperbola and their equations are respectively. (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. (vi) Eccentricity For the hyperbola we have

Definition of Special Points Lines of the Equation of Hyperbola (vii) Ordinate and Double ordinate (viii) Latus rectum (ix) Focal Distance of a Point SP = ex – aS´P = ex + a “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.”

Conjugate Hyperbola The conjugate hyperbola of the hyperbola The eccentricity of the conjugate hyperbola is

Important Terms

Auxiliary Circle and Eccentric Angle Parametric Coordinate of Hyperbola The circle drawn on transverse axis of the hyperbola as diameter is called an auxiliary circle of the hyperbola. If is the equation Of hyperbola, then its auxiliary circle is x 2 + y 2 = a 2 = eccentric angle are known as parametric equation of hyperbola.

Position of Point with respect to Hyperbola inside Outside

Intersection of a Line and a Hyperbola Let the equation of line is y = mx + c and equation of hyperbola is Point of intersection of line and hyperbola could be found out by solving the above two equations simultaneously.

Intersection of a Line and a Hyperbola This is a quadratic equation in x and therefore gives two values of x which may be real and distinct, coincident or imaginary. [Putting the value of y in the equation of Hyperbola]

Condition for Tangency and Equation of Tangent in Slope Form and Point of contact Given hyperbola is and given line is y = mx + c This is the required condition for tangency.

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

Equation of Tangent and Normal in Point Form Equation of tangent at any point (x 1, y 1 ) of the hyperbola is Equation of Normal at any point (x 1, y 1 ) of the hyperbola is

Equation of Tangent and Normal in Parametric Form Equation of tangent at is Equation of normal in parametric form is

Class Test

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.

Solution Let P(x, y) be any point of hyperbola. Let S(2, 2) be the focus. According to the definition of hyperbola This is the required equation of hyperbola.

Class Exercise - 2 Find the coordinates of centre, lengths of the axes, eccentricity, length of latus rectum, coordinates of foci, vertices and equation of directrices of the hyperbola

Solution The given equation can be written as

Solution contd.. Shifting the origin at (1, 2) without rotating the coordinate axes, i.e. Put x – 1 = X and y – 2 = Y The equation (i) becomes Centre: The coordinates of centre with respect to new axes are X = 0 and Y = 0. The coordinates of centre with respect to old axes are x – 1 = 0 and y – 2 = 0.

Solution contd.. x = 1, y = 2 Length of axes Length of transverse axes = 2b Length of conjugate axes = 2a Eccentricity

Solution contd.. Length of latus rectum Foci:Coordinates of foci with respect to new axes are, i.e.. Coordinates of foci with respect to old axes are (1, 5) and (1, –1). Vertices: The coordinates of vertices with respect to new axes are X = 0 and, i.e. X = 0 and

Solution contd.. The coordinates of axes with respect to old axes are x – 1 = 0, i.e. x = 1 and Vertices Directrices: The equation of directrices with respect to new axes are, i.e.. The equation of directrices with respect to old axes are, i.e. y = 3 and y = 1.

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

Solution (i) Centre of hyperbola is mid-point of vertices Equation of hyperbola is Let x – 4 = X, y + 1 = Y. Equation of hyperbola in new coordinate axes is.

Solution contd.. As per definition of hyperbola a = Distance between centre and vertices = 144 Abscissae of focus in new coordinates system is X = ae, i.e. x – 4 = 12e

Solution contd.. Equation of hyperbola is (ii)Coordinates of centre are (3, 2). Equation of hyperbola is a = Distance between vertex and centre Let x – 3 = X, y – 2 = Y. Equation (i) becomes

Solution contd.. Abscissae of focus is X = ae i.e. x – 3 = e(As a = 1) x = e = e + 3 [ Abscissae of focus = 5] = 1 (4 – 1) = 3

Class Exercise - 4 Find the equations of the tangents to the hyperbola 4x 2 – 9y 2 = 36 which are parallel to the line 5x – 3y = 2.

Solution Tangent is parallel to the given line 5x – 3y = 2 Equation of tangents

Class Exercise - 5 Find the locus of mid-point of portion of tangent intercepted between the axes for hyperbola

Solution Any tangent to the hyperbola is 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. be the middle point of AB

Solution contd..

Class Exercise - 6 Find the condition that the line lx + my + n = 0 will be normal to the hyperbola

Solution The equation of the given line is lx + my + n = 0...(ii) Equations (i) and (ii) will represent the same line if

Solution contd..

Class Exercise - 7 The curve represents (a)a hyperbola if k < 8 (b)an ellipse if k > 8 (c)a hyperbola if 8 < k < 12 (d)None of these

Solution The given equation represents hyperbola if (12 – k) (8 – k) < 0 i.e. 8 < k < 12 Hence, answer is (c).

Class Exercise - 8 If the line touches the hyperbola at the point, show that

Solution

Solution contd.. Both (i) and (ii) represent same line

Class Exercise - 9 Let where be two points on the hyperbola If (h, k) is the point of intersection of the normals at P and Q, then k is equal to (a) (b) (c) (d)

Solution

Solution contd.. From (iv) and (v) eliminating h, we get

Class Exercise - 10 Determine the equations of common tangent to the hyperbola

Solution The equation of other hyperbola can be written as If equations (i) and (ii) are the same, then

Solution contd..

Thank you