Question

If `P` is any point on ellipse with foci `S_1 & S_2` and eccentricity is `1/2` such that `/_PS_1S_2=alpha,/_PS_2S_1=beta,/_S_1PS_2=gamma` , then `cot(alpha/2), cot(gamma/2), cot(beta/2)` are in

Answer 1

We know that
`tan.(alpha)/(2)tan.(beta)/(2)=(1-e)/(1+e)=(1-(1//2))/(1+(1//2))=(1)/(3)`

In triangle ABC, we know that
`tan.(A)/(2)tan.(B)/(2)+tan.(C)/(2)tan.(B)/(2)+tan.(C)/(2)tan.(A)/(2)=1`
or `cot.(A)/(2)+cot.(B)/(2)+cot.(C)/(2)=cot.(A)/(2)cot.(B)/(2)cot.(C)/(2)=1`
For `DeltaPS_(1)S_(2)`
`cot.(alpha)/(2)+cot.(beta)/(2)+cot.(gamma)/(2)=3cot.(gamma)/(2)`
or `2cot.(gamma)/(2)=cot.(alpha)/(2)+cccot.(beta)/(2)`
Therefore, `cot,alpha//2,cotgamma//2,cotbeta//2` are in AP