Write the equation of the circle centered at \( (-5,-6) \) that passes through \( (2,17) \). Question Help: \( \square \) Video \( \square \) Message instructor Submit Question Jump to Answer

To find the equation of the circle, we first need to calculate its radius using the distance formula. The radius \( r \) can be found by the distance between the center \( (-5, -6) \) and the point \( (2, 17) \): \[ r = \sqrt{(2 - (-5))^2 + (17 - (-6))^2} = \sqrt{(2 + 5)^2 + (17 + 6)^2} = \sqrt{7^2 + 23^2} = \sqrt{49 + 529} = \sqrt{578}. \] Now, the standard equation of a circle is given by \( (x - h)^2 + (y - k)^2 = r^2 \), where \( (h, k) \) is the center. Plugging in the values we have: \[ (x + 5)^2 + (y + 6)^2 = 578. \] And there you have it! The final equation is \( (x + 5)^2 + (y + 6)^2 = 578 \). For a little added fun with this math, did you know that ancient Greeks studied circles extensively? The mathematician Euclid laid the groundwork for what we now call geometry, and his work still influences how we define circles today! Also, when dealing with circles, a common mistake is forgetting to square the radius in the final equation. Always double-check that you’re plugging the correct values and squaring those radius numbers to avoid any errors!