# The equation (x – α)^2 + (y – β)^2 = k(lx + my + n)^2 represents

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

(A) a parabola for k = (l2 + m2)-1

(B) an ellipse for 0 < k < (l2 + m2)-1

(C) a hyperbola for k > (l2 + m2)-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( l2 + m2)

If k(l2 + m2) = 1 , ‘P’ lies on parabola

If k(l2 + m2) < 1 , ‘P’ lies on ellipse

If k(l2 + m2) >1 , ‘P’ lies on hyperbola

If k = 0, ‘P’ lies on a point circle.

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