Question

The equation $\sqrt{\left({\left(x-2\right)}^2+y^2\right)}+\sqrt{\left({\left(x+2\right)}^2+y^2\right)}=4$ represents a/an

• A. circle
• B. straight lines
• C. parabola
• D. ellipse

Answer 1

The correct option is B

straight lines

Simplification of the given equation:

The given equation is $\sqrt{\left({\left(x-2\right)}^2+y^2\right)}+\sqrt{\left({\left(x+2\right)}^2+y^2\right)}=4$.

Squaring both sides,

$\begin{array}{rcl}\sqrt{{\left(x-2\right)}^2+y^2}& =& 4-\sqrt{{\left(x+2\right)}^2+y^2}\\ {\left(x-2\right)}^2+y^2& =& 16+\left({\left(x+2\right)}^2+y^2\right)-8\sqrt{{\left(x+2\right)}^2+y^2}\\ \left(x^2+4-4x+y^2\right)& =& 16+x^2+4+4x+y^2-8\sqrt{{\left(x+2\right)}^2+y^2}\\ -8x& =& 16-8\sqrt{{\left(x+2\right)}^2+y^2}\\ \left(x+2\right)& =& \sqrt{{\left(x+2\right)}^2+y^2}\end{array}$

Again squaring,

$\begin{array}{rcl}{\left(x+2\right)}^2& =& {\left(x+2\right)}^2+y^2\\ y^2& =& 0\\ y& =& 0\end{array}$

This is the equation of $x$-axis.

Hence, the equation is of a straight line and therefore option $\left(B\right)$ is correct.