A) \[\frac{\sqrt{5}-1}{\sqrt{5}+1}\]
B) \[\frac{1-\sqrt{5}}{1+\sqrt{5}}\]
C) \[\frac{1-\sqrt{7}}{1+\sqrt{7}}\]
D) \[\frac{\sqrt{7}-1}{\sqrt{7}+1}\]
Correct Answer: C
Solution :
\[x=2+r\cos \theta \] \[y=3+r\sin \theta \] \[\Rightarrow 2+r\cos \theta +3+r\sin \theta =7\] \[\Rightarrow r(\cos \theta +\sin \theta )=2\] \[\Rightarrow \sin \theta +\cos \theta =\frac{2}{r}=\frac{2}{\pm 4}=\pm \frac{1}{2}\] \[\Rightarrow 1+\sin 2\theta =\frac{1}{4}\]\[\Rightarrow \sin 2\theta =-\frac{3}{4}\] \[\Rightarrow \frac{2m}{1+{{m}^{2}}}=-\frac{3}{4}\]\[\Rightarrow 3{{m}^{2}}+8m+3=0\] \[\Rightarrow m=\frac{-4\pm \sqrt{7}}{1-7}\] \[\frac{1-\sqrt{7}}{1+\sqrt{7}}=\frac{{{\left( 1-\sqrt{7} \right)}^{2}}}{1-7}=\frac{8-2\sqrt{7}}{-6}=\frac{-4+\sqrt{7}}{3}\]
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