Determine the position of a point 𝐴 with respect to the circle 𝑁 if 𝑃 sub 𝑁 of 𝐴 equals 814.
Let’s recall first what this notation 𝑃 sub 𝑁 of 𝐴 means. If we have a circle centered at 𝑁 and a point 𝐴, then 𝑃 sub 𝑁 of 𝐴 denotes the power of point 𝐴 with respect to the circle 𝑁. It’s calculated using the formula 𝑃 sub 𝑁 of 𝐴 equals 𝐴𝑁 squared minus 𝑟 squared, that is, the square of the distance between point 𝐴 and the center of the circle minus the square of the circle’s radius.
We’re told here that 𝑃 sub 𝑁 of 𝐴 equals 814. So we know that 𝐴𝑁 squared minus 𝑟 squared is equal to 814. It follows then that 𝐴𝑁 squared is equal to 𝑟 squared plus 814 and, hence, that 𝐴𝑁 squared is greater than 𝑟 squared because we add a positive value to 𝑟 squared to give 𝐴𝑁 squared. As 𝐴𝑁 and 𝑟 are both lengths, they’re both positive, and so it follows that 𝐴𝑁 is greater than 𝑟. This means that the distance between points 𝐴 and 𝑁 is greater than the radius of the circle. So it’s greater than the distance from point 𝑁 to the circumference of the circle. We can therefore conclude that point 𝐴 is outside the circle 𝑁.
In fact, we can also recall a general result concerning the sign of the power of a point and the position of that point relative to a circle. If 𝑃 sub 𝑁 of 𝐴 is positive, then it follows that 𝐴𝑁 is greater than 𝑟, and so the point 𝐴 is outside the circle. If, however, 𝑃 sub 𝑁 of 𝐴 is negative, then 𝐴𝑁 is less than 𝑟, and so 𝐴 is inside the circle. Finally, if 𝑃 sub 𝑁 of 𝐴 is equal to zero, then 𝐴𝑁 is equal to 𝑟. So the distance between point 𝐴 and the center of the circle is the same as the distance between the center of the circle and any point on its circumference. So point 𝐴 lies on the circumference of the circle.
In this question, the value of 𝑃 sub 𝑁 of 𝐴 is 814, which is greater than zero, and so this confirms that point 𝐴 is outside the circle.