Line y=2x+3 is tangent to a circle with center (2,-3). Which of the length of the radius of the circle?

Step-by-step explanation:

Since it is tangent to the center, the perpendicular line passes through the center, (2,-3). We find the slope of the perpendicular line to be -1/2 (perpendicular to a slope of 2). We can use point slope form to find the line from the tangent line to the center:

y+3=-1/2(x-2)

We can simplify this to

y= -1/2x-2

We can then check the point of intersection:

-1/2x-2=2x+3

-5=2.5x

x=-2

we can plug this in to the first equation to get y=2(-2)+3=-1

So the point of intersection of the circle and the tangent line is (-2,-1)

Then we find the distance of this point to (2,-3) and we can use pythagorean theorem.

The radius of the circle is