Straight Lines
Straight Line
star_batch_jee_advanced_2025
Grade 11
Question:
The lines $L_1$ and $L_2$ denoted by $3x^2 + 10xy + 8y^2 + 14x + 22y + 15 = 0$ intersect at the point $P$ and have gradients $m_1$ and $m_2$ respectively. The acute angle between them is $\theta$. Which of the following relations hold good:
$m_1 + m_2 = \frac{3}{4}$
$m_1m_2 = \frac{3}{8}$
$\theta = \sin^{-1}\left(\frac{2}{5\sqrt{5}}\right)$
Sum of the abscissa and ordinate of point $P$ is $-1$.
Step-by-Step Solution
Key Concept: The angle between two lines is found using the formula relating their slopes, then converted to sine using trigonometric identities.
The two lines $(3x + 4y + 5) = 0$ and $(x + 2y + 3) = 0$ have slopes $m_1 = -\frac{3}{4}$ and $m_2 = -\frac{1}{2}$. Using the angle between lines formula $\tan\theta = \left|\frac{m_1 - m_2}{1 + m_1m_2}\right| = \left|\frac{-3/4 + 1/2}{1 + 3/8}\right| = \frac{2}{11}$, we get $\sin\theta = \frac{2}{5\sqrt{5}}$ using the identity $\sin\theta = \frac{\tan\theta}{\sqrt{1+\tan^2\theta}}$.
Correct Answer: 2,3