If θ is acute angle between the pair of lines \(x^2-7xy+12y^2=0\), then \(\frac{2\cos \theta+3\sin \theta}{4\sin \theta+5\cos \theta}=\)
Step-by-step Solution:
We know that \[ \tan \theta = \frac{2\sqrt{h^2 - ab}}{a + b} = \frac{2\sqrt{\frac{49}{4} - 12}}{1 + 12} = \frac{1}{13} \] Now, \[ \frac{2 \cos \theta + 3 \sin \theta}{4 \sin \theta + 5 \cos \theta} = \frac{2 + 3 \tan \theta}{4 \tan \theta + 5} \] \[ = \frac{2 + 3 \times \frac{1}{13}}{4 \times \frac{1}{13} + 5} = \frac{29}{69} \]