A radar is located at the point 𝐴 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 radar’s reach.
So we have been given the coordinates of the centre of a circle and the circle’s radius and we’re 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 ℎ, 𝑘 and radius 𝑟, then its equation is given by 𝑥 minus ℎ all squared plus 𝑦 minus 𝑘 all squared is equal to 𝑟 squared. All we need to do is substitute the relevant values of ℎ, 𝑘, and 𝑟. So let’s begin.
ℎ is the 𝑥-coordinate of the centre of the circle, so in this case, it’s negative nine. So we have 𝑥 minus negative nine all squared. We need to be very careful here. It’s not 𝑥 minus nine; it’s 𝑥 minus negative nine. So be very careful with the negative signs. Next, we have 𝑦 minus 𝑘 all squared. 𝑘 is the 𝑦-coordinate of the centre of the circle, so it’s negative five. So we have 𝑦 minus negative five all squared. Then, this is equal to 𝑟 squared. So the radius of our circle is 27.
We have then 𝑥 minus negative nine all squared plus 𝑦 minus negative five all squared is equal to 27 squared. Now, that’s the beginning of the equation of our circle, but we just need to neaten it up a little bit. So 𝑥 minus negative nine will become 𝑥 plus nine and 𝑦 minus negative five will become 𝑦 plus five.
We’ll 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 radar’s reach: 𝑥 plus nine all squared plus 𝑦 plus five all squared is equal to 729.