Conic Sections
Conic Section
star_batch_jee_advanced_2025
Grade None
Question:
If origin is shifted to point $\left(3, \frac{7}{2}\right)$ and the axes are rotated through an angle $\theta$ in clockwise sense so that equation of given hyperbola changes to the standard form $\frac{x'^2}{a^2} - \frac{y'^2}{b^2} = 1$, then $\theta$ is:
$\tan^{-1}\left(\frac{4}{3}\right)$
$\tan^{-1}\left(\frac{3}{4}\right)$
$\tan^{-1}\left(\frac{5}{3}\right)$
$\tan^{-1}\left(\frac{3}{5}\right)$
Step-by-Step Solution
Key Concept: The angle of rotation of a conic equals the arctangent of the slope of its transverse axis.
The slope of the transverse axis is $\frac{3}{4}$, which directly gives the angle of rotation as $\theta = \tan^{-1}\frac{3}{4}$. This angle describes how the hyperbola is oriented relative to the standard coordinate axes.
Correct Answer: 2