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 h

Question

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 C 1 ​ and C 2 ​ are 36 π and 4 π , respectively, then the length (in units) of latus rectum of H is

TestHub · Accepted Answer

Let H : a 2 x 2 ​ − b 2 y 2 ​ = 1 ( b 2 = a 2 ( e 2 − 1 ) ) ​ ∴ eq n of C 1 ​ Ar. = 36 π π a 2 = 36 π a = 6 ​ ∴ 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 π π r 2 = 4 π π a 2 ( e − 1 ) 2 = 4 π a 2 ( e + 1 ) 2 = 4 36 π ( e − 1 ) 2 = 4 π 36 ( e + 1 ) 2 = 4 e − 1 = 3 1 ​ e + 1 = 3 1 ​ e = 3 4 ​ − 3 2 ​ Not possible ​ ∴ b 2 = 36 ( 9 16 ​ − 1 ) = 28 ∴ L R = a 2 b 2 ​ = 6 2 × 28 ​ = 3 28 ​ ​

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