Question

The equation $\sqrt{(x-2{)}^2+y^2}-\sqrt{(x+2{)}^2+y^2}=5$ represents

• A. a hyperbola
• B. an ellipse
• C. line segment
• D. an empty set

Answer 1

The correct option is D an empty set

Let point A be $(2,0)$ and B be $(-2,0)$.
Let P be $(x,y)$.
LHS of the given equation is $PA-PB$.
$AB=4$
By triangle inequality,
$PA-PB<AB$
But in the given equation, $PA-PB=5$ .
$AB=4$.
This is not possible. Hence, the equation represents an empty set.