### Video Transcript

A radar is located at the point ๐ด negative nine and negative five covering a circular region with a radius of 27 length units. Determine the equation of the circle that gives the boundary of the radarโs reach.

So we have been given the coordinates of the centre of a circle and the circleโs radius and weโre asked to determine its equation. We have all the necessary information in order to be able to do this if we recall the centre radius form of the equation of a circle. If a circle has centre with coordinates โ, ๐ and radius ๐, then its equation is given by ๐ฅ minus โ all squared plus ๐ฆ minus ๐ all squared is equal to ๐ squared. All we need to do is substitute the relevant values of โ, ๐, and ๐. So letโs begin.

โ is the ๐ฅ-coordinate of the centre of the circle, so in this case, itโs negative nine. So we have ๐ฅ minus negative nine all squared. We need to be very careful here. Itโs not ๐ฅ minus nine; itโs ๐ฅ minus negative nine. So be very careful with the negative signs. Next, we have ๐ฆ minus ๐ all squared. ๐ is the ๐ฆ-coordinate of the centre of the circle, so itโs negative five. So we have ๐ฆ minus negative five all squared. Then, this is equal to ๐ squared. So the radius of our circle is 27.

We have then ๐ฅ minus negative nine all squared plus ๐ฆ minus negative five all squared is equal to 27 squared. Now, thatโs the beginning of the equation of our circle, but we just need to neaten it up a little bit. So ๐ฅ minus negative nine will become ๐ฅ plus nine and ๐ฆ minus negative five will become ๐ฆ plus five.

Weโll also evaluate 27 squared at this point, and that is 729. So here, we have the equation of the circle that gives the boundary of the radarโs reach: ๐ฅ plus nine all squared plus ๐ฆ plus five all squared is equal to 729.