PSQ is the focal chord of ellipse `(x^2)/(a^2)+(y^2)/(b^2)=1,a gt b`, then the harmonic mean of SP and SQ is:

Options

A. `b^(2)//a`
B. `a^(2)/b`
C. `2b^(2)//a`
D. `2a^(2)//b`

Answer

A

Explanation

Let P( `a cos alpha , b sin alpha)` and Q ( a cos ` beta , b sin beta)` be the extrmites of the focal chord PQ, then ,
`e cos ((alpha+beta)/(2))= cos ((alpha-beta)/(2))`
`implies e^(2) cos (( alpha +beta)/(2))= cos ^(2) ((alpha -beta)/(2))`
`implies e^(2){1+ cos ( alpha + beta (} ={1+ cos ( alpha -beta)}`
`implies e^(2)cos ( alpha + beta)=1-e^(2)+ cos ( alpha - beta)`
Now, SP= a-ae cos ` alpha and SQ=a -ae cos beta `
`therefore HM " of" SP and SQ`
`(2 SP.Sq)/(SP+sQ)`
`=(2a^(2)(1-e cos alpha )(1-e cos beta))/(2a -ae ( cos alpha + cos beta ))`
`=(2a(1-e cos alpha - e cos beta +e^(2) cos alpha cos beta))/(2-eae ( cos alpha + cos beta ))`` = ( ae {1-2e cos ""( alpha + beta)/(2) cos ""( alpha - beta)/(2) +(e^(2))/(2) cos ( alpha+beta)+(e^(2))/(2) cos ( alpha - beta)})/(2-2e cos"" ( alpha +beta)/(2) cos "" ( alpha - beta)/(2))`
`=(2a{1-2cos^(2)(alpha- beta)/(2)+(1)/(2)-(e^(2))/(2) +(1)/(2) cos ( alpha - beta) +(e^(2))/(2) cos (alpha - beta)})/(2-2 cos ^(2)( alpha - beta)/(2))`
`=(a{2-2-2cos ( alpha - beta)+1-e^(2)+ cos ( alpha - beta)+e^(2) cos ( alpha - beta)})/(1- cos ( alpha - beta))`
`=(a {1-cos ( alpha - beta )-e^(2){1- cos ( alpha - beta)}})/(a- cos ( alpha - beta))`
`= a(1-e^(2))=a xx(b^(2))/(a^(2))=(b^(2))/(a)`