Video: Finding the Equation of a Circle

Lauren McNaughten

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 we’ve been given a diagram of a circle on a coordinate grid and asked to find its equation. Let’s begin by recalling the general form of the equation of a circle. If a circle has a center with coordinates β„Ž, π‘˜ and a radius of π‘Ÿ units, then its equation is π‘₯ minus β„Ž all squared plus 𝑦 minus π‘˜ all squared is equal to π‘Ÿ squared.

In order to answer this question, we need to determine the values of β„Ž, π‘˜, and π‘Ÿ for the circle and diagram. We’ll begin by considering the center of the circle. The π‘₯- and 𝑦-coordinates have been labeled for us on the diagram. The π‘₯-coordinate is four, and the 𝑦-coordinate is negative seven. Therefore, we can deduce the values of β„Ž and π‘˜ straight away. β„Ž is equal to four and π‘˜ is equal to negative seven.

Next, let’s consider the value of π‘Ÿ, the radius of the circle. The other information that we’re given in the question is that the horizontal line 𝑦 plus 16 equals zero is a tangent to the circle.

Therefore, if we draw in a vertical line from the center of the circle 𝑀 down to this line 𝑦 plus 16 equals zero, it is a radius of the circle. In order to find the value of π‘Ÿ, we just need to look at the difference between the 𝑦-coordinates as the line is vertical. The equation 𝑦 plus 16 equals zero is equivalent to the equation 𝑦 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 𝑦 plus 16 is equal to zero are four, negative 16. The length of this line, remember, is the difference between the 𝑦-coordinates. So that’s the difference between negative seven and negative 16. π‘Ÿ is equal to negative seven minus negative 16, which is nine.

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

𝑦 minus negative seven is just equivalent to 𝑦 plus seven. So the second bracket in our equation can be replaced with 𝑦 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 π‘₯ minus four squared plus 𝑦 plus seven squared is equal to 81.

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