solution: This looks like the equation of a circle, but we have replaced $x$ with $\frac x2$. So, we suspect this equation defines an $\boxed{\text{ellipse}}$. To verify this we write \[\left(\frac x2 - 3\right)^2 = \frac 14 \left( x - 6\right)^2,\]and we see that the equation \[ \frac{\left(x - 6 \right)^2}{4} + y^2 = 10 \]is the equation of an ellipse.
prompt.content: <|im_start|>system
  You are a helpful assistant.<|im_end|>
  <|im_start|>user
  Determine if the graph of the equation below is a parabola, circle, ellipse, hyperbola, point, line, two lines, or empty.
  
  $\left(\frac x2 - 3\right)^2 + y^2 = 10$
  Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
  <|im_start|>assistant
  
prompt.role: user
extra_info.answer: \text{ellipse}
extra_info.question: Determine if the graph of the equation below is a parabola, circle, ellipse, hyperbola, point, line, two lines, or empty.
  
  $\left(\frac x2 - 3\right)^2 + y^2 = 10$
extra_info.split: test