It is given by Coordinate Geometry that the tangent to a circle at a point will have a slope that is a negative reciprocal of the slope of the circle's radius to that point. That is to say, the product of the two slopes will equal -1.
For the circle x2 + y2 = 20 with center at (0,0), write y = ±√(20 − x2). The point (x,y) = (4,-2) will satisfy y = -√(20 − x2). The slope of a radius drawn from (0,0) to (4,-2) will have a slope of (-2 − 0)/(4 − 0) or -1/2.
It then follows that the slope of the line sought will have a slope of -1/(-1/2) or 2.
Using the slope-intercept form of a line equation, y = mx + b, obtain -2 = 2(4) + b, which yields b (the y-intercept of the line sought) equal to -10.
The line required is then y = 2x + -10 or y = 2x − 10.
