Perpendicular Is The Key

The segment $PA$ is perpendicular to the segment $PB$ . Then the triangle $\triangle PAB$ is a right triangle with $\angle APB$ the right angle. By Pythagoras:

$PA^2 + PB^2 = AB^2$

The points $P, A, B$ are in the circumference and $\angle APB$ is a inscribed right angle then the segment $AB$ is the diameter of the circumference. So the length of $AB$ is twice the radius of the circumference. In general if we suppose that the radius is $r$ we have that

$AB = 2 r$ So: $AB^2 = (2r)^2 = 4r^2$

In this problem $r = 5$ then

$PA^2 + PB^2 = AB^2 = 4 \cdot 5^2 = 4 \cdot 25 = \color{#3D99F6}{\large 100}$

A triangle inscribed in a circle is a right triangle.The hypotenuse is the diameter of the circle. So $10^2=(PA)^2+(PB)^2=100$

You didn't need all that working did you knowing that APB is creating a right angle, making AB the diameter and the hypotenuse. Turn APB into a triangle then knowing the radius is 5, the diameter must be 10. Square ten to get 100 that's your answer