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Which of the following circles is the one of the equation π₯ minus three squared plus π¦ plus one squared is equal to four?
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To answer this question, we need to remember the center-radius form of the equation of a circle.
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If a circle has its center at the point with coordinates β, π and a radius of π units, then its equation in center-radius form is given by π₯ minus β squared plus π¦ minus π squared is equal to π squared.
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We can see that the structure of this is identical to the structure of the equation that weβve been given.
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So, we need to compare the two equations to determine the values of β, π, and π.
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Weβll then be able to work out the center and radius of our circle.
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And so, weβll be able to determine what it looks like when plotted.
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Comparing the first brackets then, we have π₯ minus β in the general form and π₯ minus three in our specific equation.
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So, we see that the value of β, the π₯-coordinate of the center of the circle, is three.
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Now, comparing the second brackets, we have π¦ minus π in the general form and π¦ plus one in our equation.
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So, we see that negative π is equal to one.
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To find the value of π, we need to multiply or divide both sides of this equation by negative one, which gives that π is equal to negative one.
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So, the π¦-coordinate of the center is negative one.
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Finally, comparing the right-hand sides of our equations, we have that π squared is equal to four.
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To find the value of π, we need to take the square root of each side of this equation.
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And as four is a square number, we should recognise that its square root is just the integer two.
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Now normally when we solve an equation by square rooting, we would take plus or minus the square root.
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But as π is the radius of a circle, it needs to take a positive value.
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So, weβre only going to take the positive square root of four.
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By comparing the equation of our circle with the center-radius form then, weβve found that our circle has a center at the point three, negative one and a radius of two units.
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We can now look at the graph weβve been given to determine which is the correct plot for this circle.
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Here is the point three, negative one.
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And we can see that we have two circles centered on this point, the circles C and D.
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We need to determine which of these circles has the correct radius.
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We can do this by drawing a line from this center to any point on the circumference of each circle.
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For simplicity, it makes sense to use a horizontal or vertical radius, like the one Iβve drawn here.
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The length of this line will be the difference in the π₯-coordinates of its two endpoints, which is three and one.
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And therefore, the length of this line, and the radius of the circle C, is two units.
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Just for comparisonβs sake, we can see that the radius of the larger circle, the blue circle, is four units, because its radius is the difference between the π₯-values of seven and three.
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So, the circle, which has a center at the point three, negative one and a radius of two units and is, therefore, the circle representing the equation π₯ minus three squared plus π¦ plus one squared equals four, is circle C.
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Notice that circle D represents a common misconception.
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It has the correct center but an incorrect radius.
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In fact, the radius thatβs been drawn is the value we would get if we had forgotten to square root.
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We know that the radius squared is four, but the radius itself is two.
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This is a common misconception or a common mistake to make.
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The correct answer, when we remember to take the square root of the value on the right-hand side of the circle given in center-radius form, is circle C.