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if x and y are the solutions of 12 sinx + 5 cos x= 2 y^2-8y+21, the value of 3864 cot(xy/2) is?

if x and y are the solutions of 12 sinx + 5 cos x= 2 y^2-8y+21, the value of 3864 cot(xy/2) is?

Grade:12

1 Answers

Rinkoo Gupta
askIITians Faculty 80 Points
8 years ago
LHS=12 sinx+5cosx</ sqrt(144+25)
=12sinx+5cosx</13
LHS</13
RHS=2(y^2-4y)+21
=2(y^2-4y+4)-8+21
=2(y-2)^2+13
hence RHS>/13
Roots of the equation exist if LHS=RHS=13
=> 2y^2-8y+21=13
=>2y^2-8y+8=0
=>y^2-4y+4=0
=>(y-2)^2=0
=>y=2
12sinx+5cosx=13
(12/13)sinx+(5/13)cosx=1
cos@sinx+sin@cosx=1 where @=arctan(5/12)
sin(x+@)=1=sin(pi/2)
x+arctan(5/12)=pi/2
x=pi/2-arctan(5/12)=arccot(5/12)
cotx=5/12
Hence 3864 cot(xy/2)=3864 cotx
=3864(5/12)=1610

Thanks & Regards
Rinkoo Gupta
AskIITians Faculty

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