Question

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

• A. circle
• B. pair of a lines
• C. a parabola
• D. an ellipse

Answer 1

The correct option is B pair of a lines

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

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

$L+M=4$ .....(1)

$L^2-M^2=-8x$

Dividing $L+M$

$L-M=-2x$ .....(2)

Adding, (1) and (2),

we get $2L=-2(x-2)$

squaring, $L^2=(x-2{)}^2$

or $(x-2{)}^2+y^2=(x-2{)}^2$

$\therefore y^2=0$
Thus the equation represents a pair of coincident lines.
Actually the given equation represents the line segment.

$y=0,-2\le x\le 2$, since the other points on $y=0$ do not satisfy the given equation.