let the equation of tangent be

y = mx + b, where m is the slope, b is the y-intercept

since the angle of the line and the x-axis is 45 degrees m = tan 45 follows m = 1

y = x + b

now we need to find b

click here to see the graph of the circle

Click to see all the steps

using the Pythagorean theorem we find the coordinates of the point T(t1, t2) where the line is touching the circle

t1 = - 5*(cos(45))

t2 = 5*(sin(45))

now we have

5*(sin(45)) = - 5*(cos(45)) + b

since sin 45 = cos 45

b = 10 (sin(45))

finally the equation of one of the tangents is

y = x + 10 (sin(45)) this is the equation of the tangent that is touching the circle in the second quadrant

the equation of the tangent that is touching the circle in the forth quadrant is

y = x - 10 (sin(45))