Question

equation √(x-2)^2+y^2 +√(x+2)^2+y^2 =4 represnts​

Answer 1

Question

What does the equation $\sqrt{(x-2)^{2}+y^{2}} +\sqrt{(x+2)^{2}+y^{2}} =4$ represent?

Required Methods

• Coordinate Geometry- Distance Formula

The distance formula finds the distance that two points are apart.

Solution

❶ Distance Formula

Let us consider the distance formula.

$\implies d=\sqrt{(x_{1}-x_{2})^{2}+(y_{1}-y_{2})^{2}}$

We can get the idea of solving the problem here.

① $x_{1}=x$

② $x_{2}=2$

③ $y_{1}=y$

④ $y_{2}=0$

This represents the distance from $(x,y)$ to $(2,0)$.

In another square root, we get the following.

① $x_{1}=x$

② $x_{2}=-2$

③ $y_{1}=y$

④ $y_{2}=0$

This represents the distance from $(x,y)$ to $(-2,0)$.

❷ Applying Distance Formula

Let's see part by part.

① $\sqrt{(x-2)^{2}+y^{2}}$ represents the distance from $(x,y)$ to $(2,0)$.

② $\sqrt{(x+2)^{2}+y^{2}}$ represents the distance from $(x,y)$ to $(-2,0)$.

③ The sum of the distances is equal to 4.

❸ Final Result

However, we observe that $(2,0)$ and $(-2,0)$ are apart by a distance of 4. 4 is the least distance since a segment that connects two points has the least distance.

So, what will be the conclusion?

Let $P(x,y)$, $A(2,0)$, $B(-2,0)$.

$\implies \overline{AP}+\overline{BP}\geq \overline{AB}=4$, equals where $P$ lies on the line segment $\overline{AB}$.

$\implies \overline{AP}+\overline{BP}=4$ so $P$ lies on the line segment $\overline{AB}$.

❹ Solution

The line segment $y=0\ (-2\leq x\leq 2)$ will be the answer.