The complete set of real values of  ‘a’ such that  the point P(a, sina) lies inside the triangle formed by the lines x - 2y + 2 = 0 ; x + y = 0 and x - y - ∏ = 0, is : 

Asked by yeeshuraj3 | 25th Jul, 2016, 04:12: PM

Expert Answer:

 
 
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F o r space t h i s space q u e s t i o n space i n space o r d e r space t o space f i n d space t h e space v a l u e space o f space a space w e space w i l l space f i r s t space f i n d space t h e space i n t e r s e c t i o n space o f space a l l space t h e s e space t h r e e space l i n e s
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 l i n e s colon
x minus 2 equals 2 y space a n d space x plus y equals 0
x plus y equals 0 rightwards double arrow x equals negative y space
s u b s i t u t e space t h e space v a l u e space o f space x space f o u n d space a b o v e space i n space t h e space e q u a t i o n space x minus 2 equals 2 y space t o space f i n d space t h e space i n t r s e c t i o n space w e space g e t space minus y minus 2 equals 2 y rightwards double arrow y equals fraction numerator negative 2 over denominator 3 end fraction a n d space h e n c e space x equals 2 over 3 s i m l i r l y
w e space c a n space f i n d space a l l space t h e space i n t e r s e c t i o n space p o i n t s space a s space s h o w n space i n space t h e space g r a p h space t h e n space a c c o r d i n g space t o space t h e space q u e s t i o n space t h e space p o i n t space m u s t space l i e space i n s i d e space t h e space t r i a n g l e space t h u s space f o r m e d space s o space w e space w i l l space
t a k e space t h e space c a s e space w h e r e space i t space l i e s space o n space t h e space t r i a n g l e space h e n c e space s a t i s f y i n g space a l l space t h e space t h r e e space p o i n t s space left parenthesis fraction numerator negative 2 over denominator 3 end fraction comma 2 over 3 right parenthesis comma left parenthesis fraction numerator p i over denominator 2 end fraction comma negative fraction numerator p i over denominator 2 end fraction right parenthesis space a n d space left parenthesis 2 p i plus 2 comma p i plus 2 right parenthesis
N o w space p o i n t space P space w i l l space s a t i s f y space a l l space t h e s e space p o i n t s.
a s space w e space k n o w space t h a t space space sin a space l i e s space b e t w e e n space minus 1 space a n d space 1 space h e n c e space w e space n e e d space t o space f i n d space a space t h a t space i s space a v a i l a b l e space f r o m space x minus c o o r d i n a t e
f r o m space f i r s t space p o i n t space w e space h a v e space a equals fraction numerator p i over denominator 2 end fraction f r o m space s e c o n d space p o i n t space a equals fraction numerator negative 2 over denominator 3 end fraction space a n d space f r o m space t h e space t h i r d space p o i n t space a equals 2 p i plus 2
s o space t h e space r a n g e space w e space c a n space c h o o s e space f o r space a space s o space t h a t space P space l i e s space i n s i d e space t h e space t r i a n g l e space i s space a space l i e s space g r e a t e r space t h a n space fraction numerator negative 2 over denominator 3 end fraction space a n d space l e s s space t h a n space 2 p i plus 2

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Answered by Vijaykumar Wani | 27th Jul, 2016, 05:46: PM