Geometry, GRE problem?

Question

Smallest distance from a point P to any point on the circle C is 5 and the largest distance from the the point P to the circle is C is 11 . If point P is situated outside the circle C, then what is the distance between centre of the circle and the point P? My view about problem: 11 is tangent to the circle C from point P Let's assume k is the point where tangent touch the circle C, p is point outside circle X radius K touch point to tangent O centre of the circle so that ko is perpendicular to Kp by right angle triangle law po2= ko2+kp2 po=5+X Ko=X Kp=11 But answer is not matching with actual answer.

Answer 1

A quick google search shows that a common way to define the largest distance from a point to a circle is the smallest distance plus the diameter of the circle; the points on the circle with the smallest and largest distance lie on the line containing P and the center of circle C.

Answer 2

 Point P is 5 units from the nearest point on the circle (call it A) and 11 units from the farthest point on the circle (call it B). Points P, A, C (center of circle), and B are collinear. The distance between A and B is 6 (11 - 5), so the distance between A and C (and B and C) is 3. The distance from P to C = 5 + 3 = 8.

Answer 3

Draw a line segment from P through the center of circle C to the opposite side of circle C. The shortest distance from P to C is where this line first intersects the circle. The furthest distance from P to C is where the line intersects the other side of the circle. The furthest point is thus one circle diameter further away than the nearest point:

Furthest distance = smallest distance + diameter

11 = 5 + diameter

6 = diameter

3 = radius

Distance from center of C to P = radius + 5 = 8 units