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Mathematics

Mathematics. Session. Hyperbola Session - 2. Session Objectives. Session Objectives. E quation of chord joining two points on the hyperbola E quation of chord whose mid-point is given E quation of pair of tangents from an external point E quation of chord of contact

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Mathematics

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  1. Mathematics

  2. Session Hyperbola Session - 2

  3. Session Objectives

  4. Session Objectives • Equation of chord joining two points on the hyperbola • Equation of chord whose mid-point is given • Equation of pair of tangents from an external point • Equation of chord of contact • Asymptotes of hyperbola • Rectangular hyperbola • Equation of rectangular hyperbola referred to its asymptotes as the axes ofcoordinates • Director circle

  5. The equation of the chord joining twopoints on the hyperbola is which reduces to Equation of the Chord Joining Two Points on the Hyperbola

  6. Equation of chord of the hyperbolawhose middle point isis given by Equation of Chord whose Mid-Point is Given

  7. Equation of pair of tangents from thepoint to the hyperbola is i.e. , where Equation of Pair of Tangents Fromand External Point

  8. Equation of chord of contact of point with respect to the hyperbolais, i.e. T = 0 Equation Chord of Contact

  9. Asymptotes of Hyperbola An asymptote is a straight line, whichmeets the conic in two points both ofwhich are situated at an infinite distance,but which is itself not altogether(entirely) at infinity.

  10. meet the given hyperbola in points,whoseabscissae are given by theequation or ... (ii) To Find the Equation of the Asymptotes of the Hyperbola Let the straight line y = mx + c ... (i)

  11. We have Hence, and c = 0 To Find the Equation of the Asymptotes of the Hyperbola If the straight line (i) be an asymptote,both roots of equation (ii) must be infinite. Hence, the coefficients of x2and x in theequation (ii) must be zero.

  12. Substituting the values of m andc iny = mx + c, we get , i.e. The combined equation of theasymptotes is To Find the Equation of the Asymptotes of the Hyperbola

  13. The bisectors of the angles between the asymptotes of the hyperbola are thecoordinate axes (or axes of thehyperbola). Points to Remember • A hyperbola and its conjugate hyperbolahave the same asymptotes. • The equation of the pair of asymptotesdiffer the hyperbola and the conjugatehyperbola by the same constant only, i.e • Hyperbola – Asymptotes = Asymptotes– Conjugate Hyperbola • The asymptotes pass through the centre of hyperbola.

  14. As we know that combined equation of asymptotes is and equation of hyperbola is Equation of pair of asymptotes and equationof hyperbola differ by a constant only. (Important) Points to Remember

  15. The equations of asymptotes of the hyperbolaare given by The angle between two asymptotes is given by Rectangular Hyperbola or Equilateral Hyperbola A hyperbola whose asymptotes are atright angles to each other is called arectangular hyperbola.

  16. If the asymptotes are at right angles,then Cor: Eccentricity of rectangular hyperbola is Rectangular Hyperbola or Equilateral Hyperbola Thus, the transverse and conjugateaxesofa rectangular hyperbola are equal.

  17. Referred to the transverse and conjugateaxes as the axes of coordinates, theequation of the rectangular hyperbola is The equation of asymptotes of the hyperbola (i) isEach of these two asymptotes is inclined at an angleof with the transverse axis. Equation of the Rectangular Hyperbola Referred to its Asymptotes as the Axesof Coordinates

  18. Now rotating the axes through an angle inclockwisedirection, keeping the origin fixed,then the axescoincide with the asymptotesof the hyperbola and Putting the values of x and y in (i), we get Equation of the Rectangular Hyperbola Referred to its Asymptotes as the Axesof Coordinates

  19. Thus, equation of rectangularhyperbola when its asymptotestaken as coordinate axes is Cor: If equation of a rectangularhyperbola be thenequation of itsconjugatehyperbola will be Equation of the Rectangular Hyperbola Referred to its Asymptotes as the Axesof Coordinates This is the transformed equationof rectangular hyperbola (i).

  20. If ‘t’ is non-zero variable, the coordinates ofany point on the rectangular hyperbolaxy = c2 can be written The point is also called point ‘t’. Parametric Form of RectangularHyperbola xy = c2

  21. The equation of the chord joining twopoints and of hyperbolaxy = c2 is Equation of Chord Joining Points ‘t1’ and ‘t2’ This is the required equation of chord.

  22. Equation of Tangent in Different Forms • Equation of tangent in point formof the hyperbola xy = c2 • Equation of tangent in parametric form

  23. Equation Normal in Different Forms • Equation of normal in point form

  24. Note: The equation of normal at is a fourthdegree equation in t. Therefore, in general four normal can be drawn from a point to the hyperbola xy = c2. Equation Normal in Parametric Forms • Equation of normal in parametricform

  25. The equations to the tangents at thepoints ‘t1’ and ‘t2’ are Coordinates of point of intersection of tangentsat ‘t1’ and ‘t2’ is . Point of Intersection of Tangents at ‘t1’ and ‘t2’ to the Hyperbola xy = c2 By solving these equations, we get point ofintersection of tangents.

  26. Director Circle The locus of intersection of tangentswhich are at right angles is calleddirector circle of Hyperbola. To find the locus of the point of intersectionof tangents which meet at right angles.

  27. Let (h, k) be their point of intersection.We have Director Circle [By putting the value of (h, k) in equations (iii) and (iv)] If between (iv) and (v), we eliminate m, we shall have a relation between h and k, i.e. locus of(h, k). Squaring and adding these equations, we get

  28. Locus of (h, k) is This is the equation of director circle, whose centreis origin and radius is Director Circle

  29. Class Test

  30. If the chord through the pointson the hyperbola passesthrough a focus, prove that Class Exercise - 1

  31. The equation of the chord joiningis If it passes through the focus (ae, 0), then By componendo and dividendo, [Proved] Solution

  32. Chords of the hyperbola touch the parabola Prove thatthe locus of their middle points is thecurve Class Exercise - 2

  33. If (h, k) be the mid-point of the chord,then the equation of the chord is T= S1, If it touches the parabola y2 = 4ax, then [Condition for tangency for any liney = mx + c to the parabola] Solution

  34. Locus of (h, k) is Solution contd..

  35. Find the point of intersection oftangents drawn to the hyperbolaat the points where itis intersected by the linelx + my + n = 0. Class Exercise - 3

  36. Equation of chord of contact drawnfrom (h, k) to the hyperbola is Equations (i) and (ii) represent same line Solution Let (h, k) be the required point. T = 0 The given line islx + my + n = 0 ... (ii)

  37. Coordinates of the required point Solution contd..

  38. Prove that the product of theperpendiculars from any pointon the hyperbola to itsasymptotesis equal to Class Exercise - 4

  39. Let be any point onthe hyperbola The equation of the asymptotes of thegiven hyperbola are Length of perpendicular from Solution

  40. Length of perpendicular from Solution contd…

  41. Class Exercise - 5 The asymptotes of a hyperbola areparallel to lines 2x + 3y = 0 and3x + 2y = 0. The hyperbola hasits centre at (1, 2) and it passesthrough (5, 3). Find its equation.

  42. Asymptotes are parallel to lines2x + 3y = 0 and3x + 2y = 0 Equations of asymptotes are 2x + 3y + k1 = 0 and 3x + 2y + k2 = 0 Solution As we know that asymptotes passes through the centre of the hyperbola.Here centre of hyperbola is (1, 2).

  43. The equations of asymptotes are 2x + 3y – 8 = 0 and 3x + 2y – 7 = 0 Equation of hyperbola is (2x + 3y – 8) (3x + 2y – 7) + c = 0 It passes through (5, 3). Equation of hyperbola is (2x + 3y – 8)(3x + 2y – 7) – 154 = 0 i.e. 6x2 + 13xy + 6y2 – 38x – 37y – 98 = 0 Solution contd..

  44. Class Exercise - 6 The chord PP´ of a rectangularhyperbola meets asymptotes inQ and Q´. Show QP = P´Q´.

  45. Equation of chord PP´ is Solution Let equation of rectangular hyperbolais xy = c2. It meets asymptotes, i.e. axes at Q and Q´respectively.

  46. [Proved] Solution contd..

  47. Class Exercise - 7 The normal at the three points P, Q, Ron a rectangular hyperbola, intersectat a point S on the curve. Prove thatcentre of the hyperbola is the centroidof PQR.

  48. It passes through a point S on the hyperbola.Let coordinates of point Solution Let equation of the rectangular hyperbolais xy = c2. Let ‘t’ be the parameter of any of points P, Q, R so that normal is ... (i)

  49. (Remember this result) This is a cubic equation in t, and gives us theparameters of the three points P, Q, R, say If (h, k) is the centroid of , Solution contd..

  50. Solution contd.. Hence, centroid is (0, 0) which is centre of the hyperbola.

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