Test 1 Sec4 #24 help!

For this one, you’re looking for the equation of a circle with a center of (0,4) that passes through \left(\dfrac{4}{3},5\right). The best way to go here is just to build the equation.

For this test, you should know the standard circle equation: A circle with center (h, k) and radius r has the equation (x-h)^2+(y-k)^2=r^2. You can fill in the left side immediately: The center of this circle is (0, 4), so the left side should be (x-0)^2+(y-4)^2=x^2+(y-4)^2.

To fill in the right side, you need to find r (or r^2). Easy enough:


\left(\dfrac{4}{3}\right)^2+1^2=r^2\\ \dfrac{16}{9}+1=r^2\\ \dfrac{16}{9}+\dfrac{9}{9}=r^2\\ \dfrac{25}{9}=r^2

So the answer you want is A.

A note for those who like quick and dirty solutions: Once you’ve got the left side of the equation, you know the answer must be either A or C. If you’re paying close attention, you’ll notice that the only difference between A and C is on the right side of the equation: A says r^2=\dfrac{25}{9} and C says r^2=\dfrac{5}{3}. If you’re in a hurry, that’s probably good enough—you know you’re looking for a squared value, and A provides it.

Another note: if you can’t remember the circle equation, you can still get this question right! Use your calculator to plug x=\dfrac{4}{3} and y=5 into each answer choice. Only one will work. 🙂

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