Conic Sections
Conic Section
Allen Star Batch
Grade 11
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: To transform a hyperbola to standard form, identify the slope of its transverse axis. The rotation angle θ equals the angle whose tangent is this slope. For a hyperbola with transverse axis slope m, we have θ = tan⁻¹(m), where clockwise rotation aligns the transverse axis with the x'-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