Consider a hyperbola H having centre at the origin and foci on the x -axis. Let C ₁ be the circle touching the hyperbola H and having the centre at the origin. Let C ₂ be the circle touching the hyperbola H at its vertex and having the centre at one of its foci. If areas (in sq units) of C₁ and C₂ are 36 and 4 , respectively, then the length (in units) of latus rectum of H is

Step-by-step solution

Let H : x ^2 a ^2 - y ^2 ~b ^2 =1 ( ~b ^2= a ^2 ( e ^2-1 ) ) aligned & eq ^ n of C ₁ & Ar. =36 & a ^2=36 & a =6 aligned eq ^ n of C ₁= x ^2+ y ^2= a ^2 Now radius of C ₂ can be a ( e -1) or a ( e +1) for r = a ( e -1) for r = a ( e +1) Ar. =4 r^2=4 a ^2( e -1)^2=4 a ^2( e +1)^2=436 (e-1)^2=4 36(e+1)^2=4 e -1= 1 3 e +1= 1 3 e = 4 3 - 2 3 Not possible aligned & b ^2=36 ( 16 9 -1 )=28 & L R= 2 ~b ^2 a = 2 28 6 = 28 3 aligned