Question Video: Finding the Distance between Three Points and the Centre of a Circle with a Known Radius given Each Point’s Power with Respect to the Circle Mathematics

Video Transcript

The power of the points 𝐴, 𝐵, and 𝐶 with respect to the circle 𝐾 are 𝑃 sub 𝐾 of 𝐴 equals four, 𝑃 sub 𝐾 of 𝐵 equals 14, and 𝑃 sub 𝐾 of 𝐶 equals negative one. The circle 𝐾 has center 𝑀 and a radius of 10 centimeters. Calculate the distance between 𝑀 and each of the points.

Let’s recall first how we calculate the power of a point with respect to a circle. For a circle 𝐾 centered at 𝑀 with a radius of 𝑟 units, the power of a point 𝐴 with respect to this circle is given by 𝑃 sub 𝐾 of 𝐴 equals 𝐴𝑀 squared minus 𝑟 squared, that is, the square of the distance between point 𝐴 and center of the circle minus the square of the radius. We’ve been given the power of three points 𝐴, 𝐵, and 𝐶 with respect to this circle 𝐾, which has a radius of 10 centimeters. Let’s consider point 𝐴 first and substitute what we know into this formula.

The power of point 𝐴 with respect to this circle is four and the radius is 10. So we have the equation four equals 𝐴𝑀 squared minus 10 squared. That’s four equals 𝐴𝑀 squared minus 100. And then we can add 100 to each side of this equation, and we find that 𝐴𝑀 squared is equal to 104. 𝐴𝑀 is therefore the square root of 104. And we take only the positive value here as 𝐴𝑀 is a length. To simplify this surd or radical, we look for square factors of 104. And we find that 104 is equal to four times 26. 𝐴𝑀 is therefore the square root of four times 26. That’s the square root of four multiplied by the square root of 26, which is two root 26. So by rearranging the power of a point formula, we found the distance between point 𝐴 and the center of the circle.

Let’s now repeat this process for points 𝐵 and 𝐶. The power of point 𝐵 with respect to the circle is 14, so we have the equation 14 is equal to 𝐵𝑀 squared minus 10 squared. That gives 14 equals 𝐵𝑀 squared minus 100. And adding 100 to each side, we find that 𝐵𝑀 squared is equal to 114. 𝐵𝑀 is then equal to the square root of 114. And as 114 has no square factors other than one, this value can’t be simplified.

Finally, let’s consider point 𝐶. The power of point 𝐶 with respect to the circle is negative one. So we have the equation negative one equals 𝐶𝑀 squared minus 10 squared. This can be rearranged to 99 is equal to 𝐶𝑀 squared. And then taking the square root of each side of this equation, we find that 𝐶𝑀 is equal to the square root of 99. Looking for square factors of 99, we recall that 99 is equal to nine multiplied by 11. So 𝐶𝑀 is the square root of nine times 11, which is the square root of nine times the square root of 11, and this is equal to three root 11.

We’ve now found the distance between 𝑀 and each of these three points. 𝐴𝑀 is equal to two root 26 centimeters. 𝐵𝑀 is equal to the square root of 114 centimeters. And 𝐶𝑀 is equal to three root 11 centimeters. We can also deduce from the sign of the power of each point its position in relation to the circle. In general, if the power of a point 𝐴 with respect to a circle 𝐾 is positive, the point lies outside the circle. If the power of the point 𝐴 with respect to circle 𝐾 is equal to zero, then the point lies on the circumference of the circle, whereas if the power of point 𝐴 with respect to circle 𝐾 is negative, the point lies inside the circle.

As the power of points 𝐴 and 𝐵 is four and 14, which are both positive, these two points lie outside the circle, whereas as the power of point 𝐶 is negative one, which is obviously negative, this point lies inside the circle. We can further confirm this if we evaluate the length of 𝐴𝑀, 𝐵𝑀, and 𝐶𝑀 as decimals. To one decimal place, they are 10.2, 10.7, and 9.9. As the lengths of 𝐴𝑀 and 𝐵𝑀 are each greater than 10, which is the radius of the circle, this confirms that points 𝐴 and 𝐵 are each outside the circle, whereas as the length of 𝐶𝑀 is less than 10, this confirms that point 𝐶 is inside the circle. This is helpful for our understanding, but it wasn’t actually required in this question.

Our answer to the problem is that 𝐴𝑀 is equal to two root 26 centimeters. 𝐵𝑀 is equal to the square root of 114 centimeters. And 𝐶𝑀 is equal to three root 11 centimeters.