1.If the angle of elevation of the a cloud from a point h meter above the lake is alpha and the depression of its reflection in the lake is beta , prove that the height of the cloud is h{1+r/1-r}m , where r=tan alpha/tan beta .

Asked by tanishqa.rathod | 16th Dec, 2015, 04:32: PM

Expert Answer:

C o n s i d e r space t h e space f o l l o w i n g space f i g u r e.
L e t space apostrophe alpha apostrophe space b e space t h e space a n g l e space o f space e l e v a t i o n space o f space t h e space c l o u d space f r o m space t h e space p o i n t space B. L e t space apostrophe h apostrophe space b e space t h e space h e i g h t space o f space t h e space p o i n t space o f space o b s e r v a t i o n space B space a b o v e space t h e space l a k e space l e v e l space B C. L e t space apostrophe beta apostrophe space b e space t h e space a n g l e space o f space d e p r e s s i o n space o f space i t s space r e f l e c t i o n space i n space t h e space l a k e. C o n s i d e r space t h e space t r i a n g l e space A B C colon tan alpha equals fraction numerator o p p over denominator a d j end fraction equals fraction numerator A C over denominator B C end fraction L e t space A C equals x space m e t r e space a n d space B C equals y space m e t r e T h e r e f o r e comma space tan alpha equals x over y rightwards double arrow y equals fraction numerator x over denominator tan alpha end fraction space space space space space space equals x c o t alpha space space space space... left parenthesis 1 right parenthesis C o n s i d e r space t h e space t r i a n g l e comma space B C F colon W e space k n o w space t h a t space h e i g h t space o f space t h e space c l o u d space f r o m space t h e space l a k e equals A C plus C E equals E F equals open parentheses x plus h close parentheses space m e t r e s tan beta equals fraction numerator o p p over denominator a d j end fraction equals fraction numerator C F over denominator B C end fraction space space space space space space space equals fraction numerator h plus x plus h over denominator y end fraction space space space space space space space equals fraction numerator x plus 2 h over denominator y end fraction space space space space... left parenthesis 2 right parenthesis S u b s t i t u t i n g space t h e space v a l u e space o f space y space i n space e q u a t i o n space left parenthesis 2 right parenthesis space 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 t a n beta equals fraction numerator x plus 2 h over denominator x c o t alpha end fraction rightwards double arrow x c o t alpha cross times tan beta equals x plus 2 h rightwards double arrow 2 h equals x c o t alpha cross times t a n beta minus x rightwards double arrow 2 h equals x open parentheses fraction numerator tan beta over denominator t a n alpha end fraction minus 1 close parentheses rightwards double arrow 2 h equals x open parentheses fraction numerator t a n beta minus tan alpha over denominator t a n alpha end fraction close parentheses rightwards double arrow x equals fraction numerator 2 h tan alpha over denominator t a n beta minus t a n alpha end fraction... left parenthesis 3 right parenthesis H e i g h t space o f space t h e space c l o u d space i s equals x plus h space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals fraction numerator 2 h t a n alpha over denominator t a n beta minus t a n alpha end fraction plus h space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals fraction numerator 2 h t a n alpha plus h t a n beta minus h t a n alpha over denominator t a n beta minus t a n alpha end fraction space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals fraction numerator h t a n alpha plus h t a n beta over denominator t a n beta minus t a n alpha end fraction space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals fraction numerator h open parentheses tan alpha plus tan beta close parentheses over denominator t a n beta minus t a n alpha end fraction space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals fraction numerator h tan beta open parentheses 1 plus begin display style fraction numerator tan alpha over denominator tan beta end fraction end style close parentheses over denominator t a n beta open parentheses 1 minus fraction numerator t a n alpha over denominator t a n beta end fraction close parentheses end fraction space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals fraction numerator h open parentheses 1 plus begin display style fraction numerator t a n alpha over denominator t a n beta end fraction end style close parentheses over denominator open parentheses 1 minus fraction numerator t a n alpha over denominator t a n beta end fraction close parentheses end fraction space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals fraction numerator h open parentheses 1 plus r close parentheses over denominator open parentheses 1 minus r close parentheses end fraction comma space w h e r e comma space r equals fraction numerator t a n alpha over denominator t a n beta end fraction space space space space space space space space space space space space space space space space space space

Answered by Vimala Ramamurthy | 17th Dec, 2015, 09:30: AM

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