Question

Let A and B be two fixed points and P be another point on 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

• A. a circle if $k_1=0$ and $k_2,k_3>0$
• B. a circle if $k_1>0,k_2<0$ and $k_3=0$
• C. an ellipse if $k_1=k_2>0$ and $k_3>0$
• D. a hyperbola if $k_2=-1$ and $k_1,k_3>0$

Answer 1

Answer: (A), (B), (C)

The correct options are
A a circle if $k_1=0$ and $k_2,k_3>0$
B a circle if $k_1>0,k_2<0$ and $k_3=0$
C an ellipse if $k_1=k_2>0$ and $k_3>0$

If $k_1=0$, the $k_{1}PA+k_{2}PB=k_3$
$\Rightarrow PB=\frac{k_3}{k_2}>0$
$\Rightarrow$ $P$ describes a circle with $B$ as centre and radius=$\frac{k_3}{k_2}$
If $k_3>0$, then
$k_{1}PA+k_{2}PB=0$
$\Rightarrow \frac{PA}{PB}=\frac{k_2}{k_1}=k>0$
$\Rightarrow$ $P$ describes a circle with $P_1{P}_2$ as its diameter $P_1,P_2$ being the points whcih
divide $AB$ internally and externally in the ratio $k:1$
If $k_1=k_2>0,k_3>0$, then
$PA+PB=\frac{k_3}{k_1}=k>0$
$\Rightarrow P$ describes an ellipse with $A$ and $B$ as its foci
if $k_2=-1$ and $k_1,k_3>0$
$k_{1}PA-PB=k_3$
$\Rightarrow P$ describes a hyperbola with $A$ and $B$ as its foci.