Question

The equation $\sqrt{{(x-2)}^2+y^2}+\sqrt{{(x+2)}^2+y^2}=4$ represents

• A. pair of lines
• B. a parabola
• C. a hyperbola
• D. none of these

Answer 1

Answer: (D)

none of these

We have, $\sqrt{{(x-2)}^2+y^2}+\sqrt{{(x-2)}^2+y^2}=4$

or, $\sqrt{{(x-2)}^2+y^2}=4-\sqrt{{(x+2)}^2+y^2}$

Squaring both the sides, we get

${(x-2)}^2+y^2=16+{(x+2)}^2+y^2-8\sqrt{{(x+2)}^2+y^2}$

$\Rightarrow \sqrt{{(x+2)}^2+y^2}=x+2.$

Squaring again, we get $y^2=0,$ which represents two coincident lines $y=0.$

But, if we substitute $y=0$ in the given equation , we get

$\sqrt{{(x-2)}^2}+\sqrt{{(x+2)}^2}=4$

which is satisfied only if $x=2$ or $x=-2.$

$\therefore$ Only $(2,0)$ and $(-2,0)$ on x-axis satisfy the given equation and not the line $y=0.$