The equation `(x-alpha)^2+(y-beta)^2=k(l x+m y+n)^2` represents a parabola for `k<(l^2+m^2)^(-1)` an ellipse for `0(1^2+m^2)^(-1)` a point circle for `k=0`

Options

A. a parabola for `k lt(l^(2)+m^(2))^(-1)`
B. an ellipse for `0 lt k lt(l^(2)+m^(2))^(-1)`
C. a hyperbola for `k gt (l^(2)+m^(2))^(-1)`
D. a point circle for k = 0

Answer

B, C, D

Explanation

`(x-alpha)^(2)+(gamma-beta)^(2)=k(lx+my+n)^(2)`
`"or "sqrt((x-alpha)^(2)+(y-beta)^(2))=sqrtksqrt(l^(2)+m^(2))((lx+my+n))/(sqrt(l^(2)+m^(2)))`
`"or "(PS)/(PM)=sqrtksqrt(l^(2)+m^(2))`
where P(x, y) is any point on the curve.
Fixed `S(alpha, beta)` is focus and fixed line `lx+my+n=0` is directrix.
If `k(l^(2)+m^(2))=1, P` lies on a parabola.
If `k(l^(2)+m^(2))lt1`, P lies on an ellipse.
If `k(l^(2)+m^(2))gt1`, P lies on a hyperbola.
If k = 0, P lies on a point circle.