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If \mathrm{\alpha, \beta, \gamma} are the eccentric angles of three points on the ellipse \mathrm{\frac{x^2}{a^2}+\frac{y^2}{b^2}=1} such that normals are concurrent, then

\mathrm{ \sin (\alpha+\beta)+\sin (\beta+\gamma)+\sin (\gamma+\alpha)=k } where \mathrm{ k= }



 

Option: 1

0


Option: 2

1


Option: 3

2


Option: 4

-1


Answers (1)

best_answer

By using solution of the above solved example and taking

\mathrm{\theta_1=\alpha, \theta_2=\beta, \theta_3=\gamma \: and\: \theta_4=\delta}, we have

\mathrm{ \sum \tan \frac{\alpha}{2} \tan \frac{\beta}{2}=0 }     .........(1)

\mathrm{\tan \frac{\alpha}{2} \tan \frac{\beta}{2} \tan \frac{\gamma}{2} \tan \frac{\delta}{2}=-1 }        ........(2)

\mathrm{ \therefore \quad t_1 t_2+t_2 t_3+t_3 t_1=-t_4\left(t_1+t_2+t_3\right) \text { by equation (1) } }

\mathrm{ \text { Or } \quad t_1 t_2+t_2 t_3+t_3 t_1\left.=\frac{-t_4\left(t_1+t_2+t_3\right)}{t_1 t_2 t_3 t_4} \text { (as } t_1 t_2 t_3 t_4=-1\right) }

\mathrm{ =\frac{t_1+t_2+t_3}{t_1 t_2 t_3}}

\mathrm{ =\frac{1}{ t_2 t_3}+\frac{1}{ t_3 t_1}+\frac{1}{ t_1 t_2}}

\mathrm{ \Rightarrow \quad \tan \frac{\alpha}{2} \tan \frac{\beta}{2}+\tan \frac{\beta}{2} \tan \frac{\gamma}{2}+\tan \frac{\gamma}{2} \tan \frac{\alpha}{2} }

\mathrm{ =\cot \frac{\alpha}{2} \cot \frac{\beta}{2}+\cot \frac{\beta}{2} \cot \frac{\gamma}{2}+\cot \frac{\gamma}{2} \cot \frac{\alpha}{2} }

\mathrm{ \Rightarrow \quad \sum\left(\tan \frac{\alpha}{2} \tan \frac{\beta}{2}-\cot \frac{\alpha}{2} \cot \frac{\beta}{2}\right)=0 }

\mathrm{ \Rightarrow \sum\left(\frac{\sin ^2 \frac{\alpha}{2} \sin ^2 \frac{\beta}{2}-\cos ^2 \frac{\alpha}{2} \cos ^2 \frac{\beta}{2}}{\sin \frac{\alpha}{2} \sin \frac{\beta}{2} \cos \frac{\alpha}{2} \cos \frac{\beta}{2}}\right)=0 }

\mathrm{ \text { Or } \sum \frac{-4 \cos \frac{\alpha+\beta}{2} \cos \frac{\alpha-\beta}{2}}{\sin \alpha \sin \beta}=0 }

\mathrm{ \text { Or } \quad \sum \frac{2(\cos \alpha+\cos \beta)}{\sin \alpha \sin \beta}=0 }

\mathrm{ \Rightarrow \quad \frac{2 \sum \sin \gamma(\cos \alpha+\cos \beta)}{\sin \alpha \sin \beta \sin \gamma}=0 }

\mathrm{ \text { Or } \sin \gamma(\cos \alpha+\cos \beta)+\sin \beta(\cos \alpha+\cos \gamma)+\sin \alpha(\cos \beta+\cos \gamma)=0 }

\mathrm{ \text { Or } \sin (\alpha+\beta)+\sin (\beta+\gamma)+\sin (\gamma+\alpha)=0 }

This is a standard property for three conormal points of the ellipse.

Hence option 1 is correct.





 

Posted by

Shailly goel

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