Given, equation of curve12−αx2+α−10y2=1for circle, 12−α=α−10⇒22=2α⇒α=11and 12−α>0⇒α<12α−10>0⇒α>10⇒ Curve represents a circle for some value of α in (10,12).Option (c) is correct.Similarly, for ellipse 12−α=α−10⇒22=2α⇒α=11Option (a) is wrong. for hyperbola one denominator must be positive and other negative∴12−α>0 and α−10<0⇒α<12 and α<10⇒ Means α<10or 12−α<0 and α−10>0⇒α>12 and α>10⇒ Means α>12Options (b) and (d) is wrong.