Question

The locus of the equation $\sqrt{{(x-2)}^2+y^2}+\sqrt{(x+2{)}^2+y^2}=4$
represent

• A. A circle
• B. A line segment
• C. A parabola
• D. An ellipse

Answer 1

Answer: (B)

A line segment

We have,

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

Now, we can write this,

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

On squaring both side and we get,

${(x-2)}^2+y^2=16+{(x+2)}^2+y^2-2\times 4\times \sqrt{{(x+2)}^2+y^2}$

$\Rightarrow x^2+4-4x+y^2=16+x^2+4+4x-8\sqrt{{(x+2)}^2+y^2}$

$\Rightarrow -4x=16+4x-8\sqrt{{(x+2)}^2+y^2}$

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

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

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

On squaring both side and we get,

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

$\Rightarrow y^2=0$

$\Rightarrow y=0$

Hence, this equation is line.