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 weβve been given a diagram of a circle on a coordinate grid and asked to find its equation. Letβs begin by recalling the general form of the equation of a circle. If a circle has a center with coordinates β, π and a radius of π units, then its equation is π₯ minus β all squared plus π¦ minus π all squared is equal to π squared.

In order to answer this question, we need to determine the values of β, π, and π for the circle and diagram. Weβll begin by considering the center of the circle. The π₯- and π¦-coordinates have been labeled for us on the diagram. The π₯-coordinate is four, and the π¦-coordinate is negative seven. Therefore, we can deduce the values of β and π straight away. β is equal to four and π is equal to negative seven.

Next, letβs consider the value of π, the radius of the circle. The other information that weβre given in the question is that the horizontal line π¦ plus 16 equals zero is a tangent to the circle.

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

Now we know the values of β, π, and π, 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 π¦ minus negative seven all squared is equal to nine squared. Weβre not going to expand the brackets in this case. But we can simplify our equation slightly.

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