Question Video: Finding the Equation of a Circle Mathematics • 11th Grade

Find the equation of the circle represented by the given figure.

03:07

Video Transcript

Find the equation of the circle represented by the given figure.

So weโ€™ve been given a diagram of a circle on a coordinate grid and asked to find its equation. Letโ€™s begin by recalling the general form of the equation of a circle. If a circle has a center with coordinates โ„Ž, ๐‘˜ and a radius of ๐‘Ÿ units, then its equation is ๐‘ฅ minus โ„Ž all squared plus ๐‘ฆ minus ๐‘˜ all squared is equal to ๐‘Ÿ squared.

In order to answer this question, we need to determine the values of โ„Ž, ๐‘˜, and ๐‘Ÿ for the circle and diagram. Weโ€™ll begin by considering the center of the circle. The ๐‘ฅ- and ๐‘ฆ-coordinates have been labeled for us on the diagram. The ๐‘ฅ-coordinate is four, and the ๐‘ฆ-coordinate is negative seven. Therefore, we can deduce the values of โ„Ž and ๐‘˜ straight away. โ„Ž is equal to four and ๐‘˜ is equal to negative seven.

Next, letโ€™s consider the value of ๐‘Ÿ, the radius of the circle. The other information that weโ€™re given in the question is that the horizontal line ๐‘ฆ plus 16 equals zero is a tangent to the circle.

Therefore, if we draw in a vertical line from the center of the circle ๐‘€ down to this line ๐‘ฆ plus 16 equals zero, it is a radius of the circle. In order to find the value of ๐‘Ÿ, we just need to look at the difference between the ๐‘ฆ-coordinates as the line is vertical. The equation ๐‘ฆ plus 16 equals zero is equivalent to the equation ๐‘ฆ is equal to negative 16, which we can see by subtracting 16 from both sides.

Therefore, the coordinates of the point where this radius meets the line ๐‘ฆ plus 16 is equal to zero are four, negative 16. The length of this line, remember, is the difference between the ๐‘ฆ-coordinates. So thatโ€™s the difference between negative seven and negative 16. ๐‘Ÿ is equal to negative seven minus negative 16, which is nine.

Now we know the values of โ„Ž, ๐‘˜, and ๐‘Ÿ, we just need to substitute them in to the general form of the equation of the circle. We have ๐‘ฅ minus four all squared plus ๐‘ฆ minus negative seven all squared is equal to nine squared. Weโ€™re not going to expand the brackets in this case. But we can simplify our equation slightly.

๐‘ฆ minus negative seven is just equivalent to ๐‘ฆ plus seven. So the second bracket in our equation can be replaced with ๐‘ฆ plus seven all squared. Equally, nine squared can be replaced with 81.

So the equation of the circle in the given figure in center radius form is ๐‘ฅ minus four squared plus ๐‘ฆ plus seven squared is equal to 81.

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