Q: The equation (x – α)^{2} + (y – β)^{2} = k(lx + my + n)^{2} represents

(A) a parabola for k = (l^{2} + m^{2})^{-1}

(B) an ellipse for 0 < k < (l^{2} + m^{2})^{-1}

(C) a hyperbola for k > (l^{2} + m^{2})^{-1}

(D) a point circle for k = 0.

Sol. (x – α)^{2} + (y – β)^{2} = k(lx + my + n)^{2}

$\large = k(l^2 + m^2)(\frac{lx + my + n}{\sqrt{l^2 + m^2}})^2 $

⇒ PS/PM = k( l^{2} + m^{2})

If k(l^{2} + m^{2}) = 1 , ‘P’ lies on parabola

If k(l^{2} + m^{2}) < 1 , ‘P’ lies on ellipse

If k(l^{2} + m^{2}) >1 , ‘P’ lies on hyperbola

If k = 0, ‘P’ lies on a point circle.

Hence (A), (B), (C) and (D) are the correct answers.