From an exterior point P that is h units from a circle of radius r, a tangent line is drawn to the circle. Let y denote the distance from the point P to the point of tangency T. Express y as a fuction of h.
To find the relation between y and h, we can use the concept of the Pythagorean theorem in geometry.
Let's consider the right triangle formed by the point P, the point of tangency T, and the center of the circle O. The radius of the circle is r, and the distance between the point P and the center of the circle O is also r. Therefore, the length of the segment OT is r.
Now, let's analyze the triangle OTY, where OT is the hypotenuse, TY is the opposite side, and OY is the adjacent side.
By the Pythagorean theorem, we know that the sum of the squares of the lengths of the two shorter sides of a right triangle is equal to the square of the length of the hypotenuse. In this case, we have:
OT^2 = TY^2 + OY^2
Substituting the known values, we get:
r^2 = y^2 + (r + h)^2
Expanding the equation, we have:
r^2 = y^2 + r^2 + 2rh + h^2
Rearranging the equation, we can solve for y:
y^2 = r^2 - r^2 - 2rh - h^2
y^2 = -2rh - h^2
Taking the square root of both sides, we get:
y = ±√(-2rh - h^2)
Therefore, the distance y from the point P to the point of tangency T as a function of h is:
y = ±√(-2rh - h^2)
Note that we have both plus and minus signs because the triangle can be both above and below the tangent line.