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 ℎ, 𝑘 and a radius of 𝑟 units, then its equation is 𝑥 minus ℎ all squared plus 𝑦 minus 𝑘 all squared is equal to 𝑟 squared.
In order to answer this question, we need to determine the values of ℎ, 𝑘, and 𝑟 for the circle and diagram. We’ll begin by considering the center of the circle. The 𝑥- and 𝑦-coordinates have been labeled for us on the diagram. The 𝑥-coordinate is four, and the 𝑦-coordinate is negative seven. Therefore, we can deduce the values of ℎ and 𝑘 straight away. ℎ is equal to four and 𝑘 is equal to negative seven.
Next, let’s consider the value of 𝑟, 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 𝑟, 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. 𝑟 is equal to negative seven minus negative 16, which is nine.
Now we know the values of ℎ, 𝑘, and 𝑟, we just need to substitute them in to the general form of the equation of the circle. We have 𝑥 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 𝑥 minus four squared plus 𝑦 plus seven squared is equal to 81.