Let the circle C touch the line x-y+1=0 , have the centre on the positive x -axis, and cut off a chord of length 4 13 along the line -3 x+2 y=1 . Let H be the hyperbola x^2 ^2 - y^2 ^2 =1 , whose one of the foci is the centre of C and the length of the transverse axis is the diameter of C . Then 2 ^2+3 ^2 is equal to ______

Step-by-step solution

r= | a+1 2 | (a+1)^2=2 r^2 Also ( 3 a-1 13 )^2+ ( 2 13 )^2=r^2 gathered ( 3 a-1 13 )^2+ 4 13 = (a+1)^2 2 5 a^2-14 a-3=0 gathered a=- 1 5 , 3 aligned & a - 1 5 & r=2 2 aligned One focus of x^2 ^2 - y^2 ^2 =1 is (3,0) aligned & e=3 and 2 =4 2 & =2 2 ^2=8 & ^2 [1+ ^2 ^2 ]=9 & ^2+ ^2=9 & ^2=1 & 2 ^2+3 ^2=19 aligned