Trig

The coordinates of the point that intersect the unit circle are (x,y)

According to the Pythagorean theorem:

x² + y² = 1²

x² + y² = 1

According to the problem statement:

y/x = -2

So,

y = -2x

Substitute:

x² + (-2x)² = 1

x² + 4x² = 1

5x² = 1

x² = 1/5

x = ±√(1/5)

x = ±√(5) / 5

y = -2[ ±√(5) / 5 ]

y = ∓2√(5) / 5

The points are: \((\frac{\sqrt5}{5},-\frac{2\sqrt5}{5})\)    and    \((-\frac{\sqrt5}{5},\frac{2\sqrt5}{5})\)

The line has the equation  y = -2x      (1)

The circle has the equation  x^2 + y^2  = 1     (2)

Sub (1)  into (2)

x^2 + (-2x)^2   = 1

x^2 + 4x^2  = 1

5x^2  =  1           divide both sides by 5

x^2  =   1/5           take both roots

x  = ± 1/√5

And using (1)   when   x  = 1/√5,   y  = -2/√5

And when  x = -1/√5,  y  =  -2 (-1/ √5)  =   2/√5

So.....the intersection points are   ( 1/√5, -2 /√5 )   and   ( -1/√5,  2/√5)