Question

The equation $|\sqrt{x^2+(y-1{)}^2}-\sqrt{x^2+(y+1{)}^2}|=k$ will represent a hyperbola for

• A. $kϵ(0,2)$
• B. $kϵ(0,1)$
• C. $kϵ(0,\infty )$
• D. $kϵ(-\infty ,2)$

Answer 1

Answer: (A), (B)

The correct options are
A

$kϵ(0,2)$

B

$kϵ(0,1)$

Equation is equivalent to $|F_{1}P-F_{2}P|$ = constant

$\therefore 2a=k$ and $2ae=F_1{F}_2=2$

$\therefore e=\frac{2}{k}$

For hyperbola, $e>1\Rightarrow \frac{2}{k}>1\Rightarrow k<2$

Also , $k>0$

$\therefore kϵ(0,2)$