Christopher J. answered • 05/29/20

Berkeley Grad Math Tutor (algebra to calculus)

Razel:

If y=x+2 is tangent to the circle with center O(2,-1), the radius of the circle is equal to the distance between the center of the circle and the line y=x+2 or y-x-2=0.

The distance between a point and a line is |ax_{0}+by_{0}+c| / √(a^{2}+b^{2})

where the line has equation ax+by+c =0 and the point is at (x_{0},y_{0})

We have y-x-2=0 and O(2,-1). So a = -1, b = 1, c = -2, x_{0 }= 2, y_{0} = -1. Plugging in those values yields

distance =5/√2.

The circle has equation (x-x_{0})^{2}+(y-y_{0})^{2} = d^{2} Plug in x_{0 }y_{0 }and d.