Question Video: Finding the Equation of a Circle when Given its Center and Diameter Mathematics • 11th Grade

Give the general form of the equation of the circle of center (8, โˆ’2) and diameter 10.

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Video Transcript

Give the general form of the equation of the circle of center eight, negative two and diameter 10.

In this question, weโ€™re given two pieces of information about our circle. It has center eight, negative two and diameter 10. We recall that the center radius form of the equation of a circle is ๐‘ฅ minus โ„Ž all squared plus ๐‘ฆ minus ๐‘˜ all squared is equal to ๐‘Ÿ squared, where the circle has center with coordinates โ„Ž, ๐‘˜ and radius ๐‘Ÿ. In our circle, we have โ„Ž equal to eight and ๐‘˜ equal to negative two. And since the diameter of the circle is equal to 10, the radius will be half of this, which is equal to five.

Substituting these values into our equation, we have ๐‘ฅ minus eight all squared plus ๐‘ฆ minus negative two all squared is equal to five squared. Simplifying ๐‘ฆ minus negative two gives us ๐‘ฆ plus two. So, our equation becomes ๐‘ฅ minus eight all squared plus ๐‘ฆ plus two all squared is equal to five squared.

Weโ€™re asked to give this equation in general form. So, we will need to distribute the parentheses. On the left-hand side, we need to multiply ๐‘ฅ minus eight by ๐‘ฅ minus eight and ๐‘ฆ plus two by ๐‘ฆ plus two. On the right-hand side, five squared is equal to 25. One way of distributing the parentheses is using the FOIL method. ๐‘ฅ minus eight multiplied by ๐‘ฅ minus eight is equal to ๐‘ฅ squared minus eight ๐‘ฅ minus eight ๐‘ฅ plus 64. In the same way, ๐‘ฆ plus two all squared is equal to ๐‘ฆ squared plus two ๐‘ฆ plus two ๐‘ฆ plus four.

Our next step is to collect like terms on the left-hand side, giving us ๐‘ฅ squared minus 16๐‘ฅ plus ๐‘ฆ squared plus four ๐‘ฆ plus 68 is equal to 25. We can then subtract 25 from both sides. And writing the quadratic terms first, we have ๐‘ฅ squared plus ๐‘ฆ squared minus 16๐‘ฅ plus four ๐‘ฆ plus 43 equals zero. This is the general form of the equation of the circle with center eight, negative two and diameter 10.

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