MCQ
Curve passing through $(3, 0)$ and satisfying the differential equation $\left( {9 - {x^2}} \right){\left( {\frac{{dy}}{{dx}}} \right)^2} = 9 - {y^2}$ represents
- AStraight line
- ✓circle
- Cparabola
- DEllipse
$\Rightarrow \sin ^{-1}\left(\frac{\mathrm{x}}{3}\right) \pm \sin ^{-1}\left(\frac{\mathrm{y}}{3}\right)=\mathrm{c}$
passes through $(3,0)$
$\Rightarrow \frac{\pi}{2} \pm 0=\mathrm{c}$
$\Rightarrow \sin ^{-1}\left(\frac{x}{3}\right)=\frac{\pi}{2} \pm \sin ^{-1}\left(\frac{y}{3}\right)$
$\Rightarrow \frac{\mathrm{x}}{3}=\sqrt{1-\frac{\mathrm{y}^{2}}{9}} \Rightarrow \mathrm{x}^{2}+\mathrm{y}^{2}=9$
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$g(\theta)=\sqrt{f(\theta)-1}+\sqrt{f\left(\frac{\pi}{2}-\theta\right)-1}$
where
$f(\theta)=\frac{1}{2}\left|\begin{array}{ccc}1 & \sin \theta & 1 \\ -\sin \theta & 1 & \sin \theta \\ -1 & -\sin \theta & 1\end{array}\right|+\left|\begin{array}{ccc}\sin \pi & \cos \left(\theta+\frac{\pi}{4}\right) & \tan \left(\theta-\frac{\pi}{4}\right) \\ \sin \left(\theta-\frac{\pi}{4}\right) & -\cos \frac{\pi}{2} & \log _e\left(\frac{4}{\pi}\right) \\ \cot \left(\theta+\frac{\pi}{4}\right) & \log _e\left(\frac{\pi}{4}\right) & \tan \pi\end{array}\right|$.
Let $p (x)$ be a quadratic polynomial whose roots are the maximum and minimum values of the function $g(\theta)$, and $p(2)=2-\sqrt{2}$. Then, which of the following is/are TRUE ?
$(A)$ $p \left(\frac{3+\sqrt{2}}{4}\right)<0$
$(B)$ $p \left(\frac{1+3 \sqrt{2}}{4}\right)>0$
$(C)$ $p \left(\frac{5 \sqrt{2}-1}{4}\right)>0$
$(D)$ $p \left(\frac{5-\sqrt{2}}{4}\right)<0$
$\frac{15}{16}$
$\frac{3}{16}$
$-\frac{3}{16}$
$-\frac{16}{3}$