Video: Finding the Equation of a Circle

A radar is located at the point ๐ด (โˆ’9, โˆ’5) covering a circular region with a radius of 27 length units. Determine the equation of the circle that gives the boundary of the radarโ€™s reach.

02:14

Video Transcript

A radar is located at the point ๐ด 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 radarโ€™s reach.

So we have been given the coordinates of the centre of a circle and the circleโ€™s radius and weโ€™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 โ„Ž, ๐‘˜ and radius ๐‘Ÿ, then its equation is given by ๐‘ฅ minus โ„Ž all squared plus ๐‘ฆ minus ๐‘˜ all squared is equal to ๐‘Ÿ squared. All we need to do is substitute the relevant values of โ„Ž, ๐‘˜, and ๐‘Ÿ. So letโ€™s begin.

โ„Ž is the ๐‘ฅ-coordinate of the centre of the circle, so in this case, itโ€™s negative nine. So we have ๐‘ฅ minus negative nine all squared. We need to be very careful here. Itโ€™s not ๐‘ฅ minus nine; itโ€™s ๐‘ฅ minus negative nine. So be very careful with the negative signs. Next, we have ๐‘ฆ minus ๐‘˜ all squared. ๐‘˜ is the ๐‘ฆ-coordinate of the centre of the circle, so itโ€™s negative five. So we have ๐‘ฆ minus negative five all squared. Then, this is equal to ๐‘Ÿ squared. So the radius of our circle is 27.

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

Weโ€™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 radarโ€™s reach: ๐‘ฅ plus nine all squared plus ๐‘ฆ plus five all squared is equal to 729.

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