T h e space f o c i space o f space t h e space e l l i p s e space x squared over 16 plus y squared over b squared equals 1 space a n d space t h e space h y p e r b o l a space x squared over 144 minus y squared over 81 equals 1 over 25 space c o i n c i d e comma t h e n space f i n d space t h e space v a l u e space o f space b squared.

Asked by surajsubedi14 | 28th Feb, 2015, 08:07: PM

Expert Answer:

C o n s i d e r space t h e space e q u a t i o n space o f space t h e space h y p e r b o l a comma space x squared over 144 minus y squared over 81 equals 1 over 25 R e w r i t i n g space t h e space a b o v e space e q u a t i o n comma space w e space h a v e comma fraction numerator 25 x squared over denominator 144 end fraction minus fraction numerator 25 y squared over denominator 81 end fraction equals 1 rightwards double arrow fraction numerator x squared over denominator begin display style 144 over 25 end style end fraction minus fraction numerator y squared over denominator begin display style 81 over 25 end style end fraction equals 1 rightwards double arrow A squared equals 144 over 25 space a n d space B squared equals 81 over 25 rightwards double arrow A equals 12 over 5 space a n d space B equals 9 over 5 F o r space h y p e r b o p l a comma space w e space k n o w space t h a t comma space B squared equals A squared open parentheses E squared minus 1 close parentheses rightwards double arrow 81 over 25 equals 144 over 25 open parentheses E squared minus 1 close parentheses rightwards double arrow 81 equals 144 open parentheses E squared minus 1 close parentheses rightwards double arrow 81 over 144 equals E squared minus 1 rightwards double arrow E squared equals 1 plus 81 over 144 rightwards double arrow E squared equals fraction numerator 144 plus 81 over denominator 144 end fraction rightwards double arrow E squared equals 225 over 144 rightwards double arrow E equals 15 over 12 T h u s comma space t h e space f o c u s space o f space h y p e r b o l a space i s space A cross times E equals 12 over 5 cross times 15 over 12 equals 3 rightwards double arrow A E equals 3 G i v e n space t h a t space t h e space f o c u s space o f space t h e space h y p e r b o l a space a n d space t h e space e l l i p s e space c o i n c i d e. T h u s comma space t h e space f o c u s space o f space e l l i p s e equals 3 rightwards double arrow a e equals 3 rightwards double arrow open parentheses a e close parentheses squared equals 3 squared rightwards double arrow a squared e squared equals 9... left parenthesis 1 right parenthesis C o n s i d e r space t h e space g i v e n space e q u a t i o n o f space t h e space e l l i p s e comma x squared over 16 plus y squared over b squared equals 1 H e r e comma space a squared equals 16 F r o m space e q u a t i o n space left parenthesis 1 right parenthesis comma space w e space h a v e comma a squared e squared equals 9 rightwards double arrow 16 e squared equals 9 rightwards double arrow e squared equals 9 over 16 W e space k n o w space t h a t comma space b squared equals a squared open parentheses 1 minus e squared close parentheses rightwards double arrow b squared equals 16 open parentheses 1 minus 9 over 16 close parentheses rightwards double arrow b squared equals 16 open parentheses fraction numerator 16 minus 9 over denominator 16 end fraction close parentheses rightwards double arrow b squared equals 16 minus 9 rightwards double arrow b squared equals 7

Answered by Vimala Ramamurthy | 1st Mar, 2015, 11:40: AM