Line tangent to a circle

Question

How would you solve an equation like this for example:

Line Z is tangent to the circle (x−4)^2+(y−3)^2=100 at point (4, 13). Line W is tangent to this circle at point B, with Line W parallel to (but distinct from) Line Z. What are the coordinates of point B?

Answer 1

If there are 2 (different) tangents to the same circle, and they are parallel, then they must be on opposite side of the circle.

"Line Z is tangent to the circle (x−4)^2+(y−3)^2=100 at point (4, 13)"

From this we know that the center of the circle is at (4,3), and the radius is 10. Since the tangent point (4,13) has the same x value as the center, line Z has a slope of zero (horizontal), and it's equation is simply y = 13.

Line W is on the opposite side of the circle and has the equation y = -7 (displaced from line Z by twice the radius of the circle)

From all this, point B is at (4,-7)

To see all of this, draw a picture