A ray of light passing through (-3,1) reflects on x-axis
at A and the reflected ray is tangential to the circle
X^2+y^2=1 at P.
A) The reflected ray is 3x-4y+5=0
B)A=(5/3,0)
C)Area of triangle APO is 2/3 square units
('0' is origin )
D)P=(-3/5,4/5)

Asked by Sunil Soni | 22nd Sep, 2015, 10:00: AM

Expert Answer:

R e f e r space t o space t h e space f i g u r e space a h o w n space a b o v e. T h e space l i n e space p a s sin g space t h r o u g h space C left parenthesis negative 3 comma space 1 right parenthesis space i s space r e f l e c t e d space a t space A space o n space X minus a x i s. A P space i s space t h e space r e f l e c t e d space r a y. space G i v e n space t h a t space A P space i s space a space tan g e n t t o space t h e space c i r c l e space x squared plus y squared equals 1. H e n c e comma space t h e space o r i g i n a l space l i n e space m u s t space a l s o space b e space a space tan g e n t space t o space t h e space c i r c l e. space F r o m space t h e space f i g u r e space w e space c a n space s e e space t w o space tan g e n t s space d r a w n space f r o m space C left parenthesis negative 3 comma space 1 right parenthesis t o space t h e space g i v e n space c i r c l e. space C D space i s space p a r a l l e l space t o space X minus a x i s. space H e n c e comma space C D equals 3. space O D equals r a d i u s space o f space t h e space c i r c l e equals 1. H e n c e comma space i n space t r a i n g l e space C O D comma space tan a equals 1 third H e n c e comma space tan 2 a equals fraction numerator 2 tan a over denominator 1 minus tan squared a end fraction equals fraction numerator 2 cross times begin display style 1 third end style over denominator 1 minus begin display style 1 over 9 end style end fraction equals 3 over 4 F r o m space t h e space f i g u r e comma space s l o p e space o f space l i n e space C E space i s space tan open parentheses pi minus 2 a close parentheses equals negative tan 2 a equals negative 3 over 4 H e n c e comma space e q u a t i o n space o f space t h e space l i n e space C E space i s space y minus 1 equals negative 3 over 4 open parentheses x plus 3 close parentheses A space i s space t h e space p o i n t space o f space i n t e r s e c t i o n space o f space a b o v e space l i n e space w i t h space X minus a x i s. space P u t space y equals 0 space i n space t h e space s t r a i g h t space l i n e space e q u a t i o n space t o space g e t space X minus c o o r d i n a t e space o f space A. x equals 4 over 3 minus 3 equals negative 5 over 3 H e n c e comma space t h e space c o o r d i n a t e s space o f space A space a r e space open parentheses negative 5 over 3 comma space 0 close parentheses... left parenthesis 1 right parenthesis I n space t r i a n g l e space A P O comma space O A equals 5 over 3 comma space O P equals 1. space H e n c e comma space A P space equals space 4 over 3 space left parenthesis u sin g space p y t h a g o r a s space t h e o r e m right parenthesis H e n c e comma space a r e a space o f space t r i a n g l e space A P O equals 1 half cross times A P cross times O P equals 1 half cross times 4 over 3 cross times 1 equals 2 over 3... left parenthesis 2 right parenthesis F r o m space t r i a n g l e space A P O comma space tan b equals 3 over 4 rightwards double arrow sin b equals 3 over 5 comma space cos b equals 4 over 5 E q u a t i o n space o f space A P space i s space y minus 0 equals 3 over 4 open parentheses x minus open parentheses negative 5 over 3 close parentheses close parentheses rightwards double arrow 4 y equals 3 x plus 5 rightwards double arrow 3 x minus 4 y plus 5 equals 0... left parenthesis 3 right parenthesis X space c o o r d i n a t e space o f space P space equals space X space c o o r d i n a t e space o f space A space plus space A P cos b equals negative 5 over 3 plus 4 over 3 cross times 4 over 5 equals negative 5 over 3 plus 16 over 15 equals negative 3 over 5 Y space c o o r d i n a t e space o f space P space equals space Y space c o o r d i n a t e space o f space A space plus space A P sin b equals 0 plus 4 over 3 cross times 3 over 5 equals 4 over 5 H e n c e comma space t h e space c o o r d i n a t e s space o f space P space a r e space open parentheses negative 3 over 5 comma space 4 over 5 close parentheses... left parenthesis 4 right parenthesis. F r o m space left parenthesis 1 right parenthesis comma space left parenthesis 2 right parenthesis comma space left parenthesis 3 right parenthesis space a n d space left parenthesis 4 right parenthesis comma space O p t i o n s space A comma space C space a n d space D space a r e space c o r r e c t.

Answered by satyajit samal | 22nd Sep, 2015, 12:21: PM