Straight Lines
Straight Line
star_batch_jee_advanced_2025
Grade 11
Question:
If the two lines represented by $x^2\left(\tan^2 \theta + \cos^2 \theta\right) - 2y \tan \theta + y^2 \sin^2 \theta = 0$ make angles $\alpha, \beta$ with the x-axis, then:
$\tan \alpha + \tan \beta = 4\cos ec 2\theta$
$\tan \alpha \tan \beta = \sec^2 \theta + \tan^2 \theta$
$\tan \alpha - \tan \beta = 2$
$\frac{\tan \alpha}{\tan \beta} = \frac{2 + \sin 2\theta}{2 - \sin 2\theta}$
Step-by-Step Solution
Key Concept: Express tangent of angles through trigonometric identities and use sum/difference formulas to find relationships.
Let the two lines be $y = x\tan\alpha$ and $y = x\tan\beta$. We compute $\tan\alpha + \tan\beta = \frac{2\tan\theta}{\sin^2\theta} - \frac{2}{\sin\theta\cos\theta} = 4\cos 2\theta$ using trigonometric identities. Then $\tan\alpha\tan\beta = \frac{\tan^2\theta + \cos^2\theta}{\sin^2\theta} = \sec^2\theta + \cot^2\theta$. We find $\tan\alpha - \tan\beta = \pm 2$ by evaluating the discriminant. Finally, $\frac{\tan\alpha}{\tan\beta} = \frac{4\cos 2\theta + 2}{4\cos 2\theta - 2} = \frac{2 + \sin 2\theta}{2 - \sin 2\theta}$.
Correct Answer: 1,3,4