A hyperbola H:x29-y24=1 ntersects the circle, C:x2+y2-8x=0 at the points A and B.Statement-1: 2x-5y+4=0 is a common tangentto both C and H. Statement-2: Circle on AB as a diameter passes through the centre of the hyperbola H.

Equation of a tangent with slope 25 x29-y24=1 is y=25x+9×45-4⇒2x-5y+4=0Next 2x-5y+4=0 touches the circle (x-4)2+y2 = 16 if the length of the perpendicular from (4, 0) on the line is 4 which is true. Hence statement-1 is true. In statement-2, let A be (3secθ,2tanθ)A lies on the circle x2+y2-8x=0⇒13sec2θ-24secθ-4=0⇒secθ=2⇒tanθ=±3So the coordinate of A are (6,23) and of B are(6,-23) and equation of the circle on AB as diameter is (x-6)(x-6)+(y-23)(y+23)=0⇒x2+y2-12x+24=0 which does not pass through the centre (0, 0) of the hyperbola. Thus statement-2 is false.