### Video Transcript

Determine the position of a point π΄ with respect to the circle π if π sub π of π΄ equals 814.

Letβs recall first what this notation π sub π of π΄ means. If we have a circle centered at π and a point π΄, then π sub π of π΄ denotes the power of point π΄ with respect to the circle π. Itβs calculated using the formula π sub π of π΄ equals π΄π squared minus π squared, that is, the square of the distance between point π΄ and the center of the circle minus the square of the circleβs radius.

Weβre told here that π sub π of π΄ equals 814. So we know that π΄π squared minus π squared is equal to 814. It follows then that π΄π squared is equal to π squared plus 814 and, hence, that π΄π squared is greater than π squared because we add a positive value to π squared to give π΄π squared. As π΄π and π are both lengths, theyβre both positive, and so it follows that π΄π is greater than π. This means that the distance between points π΄ and π is greater than the radius of the circle. So itβs greater than the distance from point π to the circumference of the circle. We can therefore conclude that point π΄ is outside the circle π.

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 π sub π of π΄ is positive, then it follows that π΄π is greater than π, and so the point π΄ is outside the circle. If, however, π sub π of π΄ is negative, then π΄π is less than π, and so π΄ is inside the circle. Finally, if π sub π of π΄ is equal to zero, then π΄π is equal to π. So the distance between point π΄ 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 π΄ lies on the circumference of the circle.

In this question, the value of π sub π of π΄ is 814, which is greater than zero, and so this confirms that point π΄ is outside the circle.