Let `A` and `B` be two fixed points and `P`, another point in the plane, moves in such a way that `k_(1)PA+k_(2)PB=k_(3)`, where `k_(1)`, `k_(2)` and `k_(3)` are real constants. The locus of `P` is
Which one of the above is not true ?

Options

A. a circle if `k_(1)=0` and `k_(2)`, `k_(3) gt 0`
B. a circle if `k_(1) gt 0` and `k_(2) lt 0`, `k_(3) = 0`
C. an ellipse if `k_(1)=k_(2) gt =0` and `k_(3) gt 0`
D. a hyperbola if `k_(2)=-1` and `k_(1)`, `k_(3) gt 0`

Explanation

If `k_(1)=0`, then
`k_(1)PA+k_(2)PB=k_(3)`
`impliesPB=(k_(3))/(k_(2)) gt 0` ,brgt `implies P` describes a circle with `B` as centre and radius `=(k_(3))/(k_(2))`
If `k_(3)=0`, then
`k_(1)PA+k_(2)PB=0`
`implies(PA)/(PB)=(k_(2))/(-k_(1))= k gt 0`
`impliesP` describes a circle with `P_(1)P_(2)` as its diameter, `P_(1)` and `P_(2)` being the points which divide `AB`. Internally and externally in the ratio `k : 1`
If `k_(1)=k_(2) gt 0` and `k_(3) gt 0`, then
`PA+PB=(k_(3))/(k_(1))=k gt 0`
`impliesP` describes an ellipse with `A` and `B` as its foci.