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 \(s\).
+128460 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
