Give the general form of the
equation of the circle of center eight, negative two and diameter 10.
In this question, we’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 𝑥 minus ℎ all squared plus 𝑦 minus 𝑘 all
squared is equal to 𝑟 squared, where the circle has center with coordinates ℎ, 𝑘
and radius 𝑟. In our circle, we have ℎ equal to
eight and 𝑘 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 𝑥 minus eight all squared plus 𝑦 minus negative two all squared
is equal to five squared. Simplifying 𝑦 minus negative two
gives us 𝑦 plus two. So, our equation becomes 𝑥 minus
eight all squared plus 𝑦 plus two all squared is equal to five squared.
We’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 𝑥 minus eight by 𝑥 minus eight and 𝑦 plus two by 𝑦 plus two. On the right-hand side, five
squared is equal to 25. One way of distributing the
parentheses is using the FOIL method. 𝑥 minus eight multiplied by 𝑥
minus eight is equal to 𝑥 squared minus eight 𝑥 minus eight 𝑥 plus 64. In the same way, 𝑦 plus two all
squared is equal to 𝑦 squared plus two 𝑦 plus two 𝑦 plus four.
Our next step is to collect like
terms on the left-hand side, giving us 𝑥 squared minus 16𝑥 plus 𝑦 squared plus
four 𝑦 plus 68 is equal to 25. We can then subtract 25 from both
sides. And writing the quadratic terms
first, we have 𝑥 squared plus 𝑦 squared minus 16𝑥 plus four 𝑦 plus 43 equals
zero. This is the general form of the
equation of the circle with center eight, negative two and diameter 10.