Asymptotes are 3x−4y−1=0⋅⋅⋅⋅⋅⋅⋅(i) 4x−3y−6=0⋅⋅⋅⋅⋅⋅⋅(ii) By Eqs. (i) and (ii), we get x=3,y=2 Now, the direction vectors of asymptotes for 3x−4y=1⇒d1=⟨4,3⟩ and for 4x−3y=6⇒d2=⟨3,4⟩ Thus, transverse and conjugate axis directions.
L1
√a12+b12
=±
L2
√a22+b22
⇒
3x−4y−1
√32+(−4)2
=±
4x−3y−6
√42+(−3)2
⇒3x−4y−1=±4x−3y−6 So, 3x−4y−1=4x−3y−6 or 3x−4y−1=−4x+3y+6 ⇒x+y−5=0 or 7x−7y−7=0 ⇒x+y−5=0 or x−y−1= Hence, transverse and conjugate axis are x+y=5,x−y=1