1. Find the equations of the tangents to the  hyperbola  9x² - 16y² = 1 drawn parallel to 9x + 8y = 10.

          

2. Find the equations of tangents to the hyperbola x²-4y² =4 drawn perpendicular to x +2y = 0.

4. Determine the equation to hyperbola whose centre at (0, 0), distance between the foci is 18 and distance between the directrices is 8.
    Sol:  Given C = (0, 0)
                Distance between foci: 18
               ⇒  2ae = 18
               ⇒   ae = 9 ......... (1)
     Distance between Directrices = 8
         ⇒   2a/e    = 8
         ⇒   a/e  = 4
     multiplying  (1)  and  (2)
             ⇒  (ae) a/e  =36
              ⇒  a² = 36
     We know that b² = a² (e² - 1)
             ⇒   b² = 81 - 36
             ⇒   b² = 45

 

6. Find the equation of the hyperbola whose centre is (1, 0), focus is (6, 0) and the length of transverse axis is 6.
Sol:      Given:  Centre: C (1, 0)
                 Focus:  S (6, 0)
                 Length of transverse axis: 2a = 6
              ⇒   a = 3
                 We know that: CS = ae = | 1- 6 | = 5
              ⇒       ae = 5
                 Also, b² = a²(e² -1)
               ⇒   b² = (ae)² - a²
               ⇒   b² = 25 - 9
               ⇒   b² = 16
                       b = 4


                   Since it represents a hyperbola
                   then 9 - C > 0 and 5 - C < 0
                 ⇒   9 > C and 5 < C
                 ⇒   C < 9 ............. (1)   and   5 < C ............. (2)
                    from (1)  and  (2)
                    5 < C < 9

8. Find the equations of the normal to the hyperbola  x² -3y² = 144  at the end of latus rectum in first quadrant.

Sol:

       a² = 144; b² = 48      (  ^..^. a > b) 

10. Find the equation of the hyperbola with eccentricity ,  focus (1, 2) and corresponding   directrix  is  2x + y = 1
Sol:     Given : focus = S(1, 2)
                 Eccentricity = e = 
                 Directrix = 2x + y- 1 = 0
                 Let P(x, y) be any point on the hyperbola
                 By definition of Hyperbola PS = ePM

            ⇒    5(x² + 1- 2x + y² + 4 - 4y) = 3(4x² + y² + 1 + 4xy - 2y- 4x)
           ⇒    12x² + 3y² + 3+12xy - 6y - 12x - 5x² - 5 +10x - 5y² - 20 + 20y = 0
            ⇒   7x² + 12xy − 2y² - 2x + 14y - 22 = 0

11. Find the centre, eccentricity, foci, vertices, directrices and length of latus rectum of the  hyperbola 9x² - 16y² + 72x - 32y - 16 = 0
Sol:    Given hyperbola = 9x² - 16y² + 72x - 32y = 16 = 0
              ⇒    9(x² + 8x) − 16 (y² + 2y) = 16

13.  Show that the locus of poles w.r.t. the parabola  y² = 4ax  of the tangents to the 
       rectangular hyperbola x² - y² = a² lies on the ellipse 4x² + y² = 4a²
Sol:    Given parabola: y² = 4ax
                  Polar of (x₁, y₁) : yy₁ = 2a(x + x₁)

      
Given rectangular hyperbola : x²- y² = a²
By data condition = c² = a²m² - b²
               c² = a²(m²-1)   [  ^..^. b = a]
         
         ⇒       4x₁² = 4a² - y₁²
         ⇒      4x₁² + y₁² = 4a²
              Locus of P (x₁, y₁) : 4x² + y² = 4a²

14.  If  e, e₁  are the eccentricities of a hyperbola and its conjugate hyperbola