Y = x+ 1- This is a straight line with an angle of 45 to the right, and the coordinate intersecting with the Y axis is (0, 1).

Make a straight line perpendicular to the origin, y =-x.

This line intersects the previous line at point H.

H(-0.5, 0.5) can be solved.

Let M(a, b) be the intersection of a straight line y=x+ 1 and a circle o.

Draw MN parallel to the y axis and HN parallel to the x axis.

As can be seen from the figure, HM=0.5 root 6.

HN=MN=0.5 2HM=0.25 12=0.5 3。

M(a, b)a=0.5 root 3-0.5? =0.5x0.732

B=0.5 root 3+0.5=0.5x2.232

mo^2=a^2+b^2=0. 133956+ 1.245456= 1.3794 12

MO = 1. 1744837 163622 150435

The equation of a circle with O as the center and R as the radius is

x^2+y^2=R^2

Where r = mo =1.1744 8371636 2215044534.

I don't know why I gave such a strange number.

In order to get the shortest tangent of the first quadrant, the tangent point should be on the diameter of circle O45, and its coordinates are (c, d), and c=d= 1.4 14 radius. The intersection of tangents on the y axis Y=2d, and the intersection of tangents on the x axis X=2c.

for reference only

Especially that strange number, please check it carefully. The way to solve the problem should be correct.