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Let \(\gamma\) be the circle of equation \(x^2 + y^2 = 4\). If \(r, s\) are two lines intersecting at point \(P = (1,3)\) and are tangent to \(\gamma\), then find the cosine of the angle between \(r\) and \(s\).

 Jan 26, 2021
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Let  the center of the  circle  be, O  = (0,0)

 

The distance  from  (0,0)  to (1,3)  = sqrt (1^2 + 3^2)  = sqrt (10)

 

And if we let  the radius of the  circle     meet one of the tangents at  B

 

Then triangle  OPB is  right     with OP  the hypotenuse and OB  a leg = the radius =  2

 

And    PB    = sqrt  ( OP^2  - OB^2)  =  sqrt (10  - 2^2)  =  sqrt (6)

 

sin OPB  =  2/sqrt (10)

 

So   ....using symmetry......the angle  between r and s    will be 2*angle OPB 

 

cos  ( 2 * angle OPB) =    1  - 2 (sin (OPB) )^2  =    1  - 2 (2/sqrt (10))^2  =

 

1 -  2  ( 4/10)  =

 

1  - 8/10  =

 

2/10   =   

 

1/5  =   cos  of the  angle  beteen  r and  s

 

 

cool cool cool

 Jan 26, 2021
edited by CPhill  Jan 26, 2021

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