Straight Lines
Straight Line
star_batch_jee_advanced_2025
Grade 11
Question:
The equation of a pair of straight lines is $ax^2 + 2hxy + by^2 = 0$. If the angle by which the axes be rotated so the term containing $xy$ in the equation may be removed is $\phi$, then:
$\phi = \frac{\pi}{8}$ if $a = b, h > 0$
$\phi = \frac{\pi}{8}$, if $2h = a - b$
$\phi = \frac{\pi}{8}$ if $a = b, h \neq 0$
$\phi = \frac{\pi}{4}$, if $2h = a - b$
Step-by-Step Solution
Key Concept: The angle of rotation that eliminates the $x'y'$ term in a pair of lines is found by setting the coefficient of $xy$ to zero.
When axes are rotated by angle $\phi$ about the origin in the anticlockwise sense, coordinates transform as $x = x'\cos\phi - y'\sin\phi$ and $y = x'\sin\phi + y'\cos\phi$. Substituting into the pair of lines equation $ax^2 + 2hxy + by^2 = 0$ and expanding yields a transformed equation. For the $xy$ term to vanish in the new coordinates, we require $(b-a)\sin 2\phi + 2h\cos 2\phi = 0$, which gives $\tan 2\phi = \frac{2h}{a-b}$, hence $\phi = \frac{1}{2}\tan^{-1}\left(\frac{2h}{a-b}\right)$.
Correct Answer: 1,2