91Ó°ÊÓ

Two rays are $r_1$and $r_2$.

The distance between two points is the length of a line that connects two points in a plane.

The formula for finding the distance between two points is usually given by,

$\mathbf{d}\mathbf{=}\sqrt{{\mathbf{(}{\mathbf{x}}_{\mathbf{2}}\mathbf{−}{\mathbf{x}}_{\mathbf{1}}\mathbf{)}}^{\mathbf{2}}\mathbf{+}{\mathbf{(}{\mathbf{y}}_{\mathbf{2}}\mathbf{−}{\mathbf{y}}_{\mathbf{1}}\mathbf{)}}^{\mathbf{2}}}$

This formula is used to find the distance between any two points in the coordinate plane or x-y plane.

Let $S_1\left(a,0\right)$and $S_2\left(−a,0\right)$.

Let the point with constant path difference is $\left(x,y\right)\left(P\right)$.

localid="1663399680029" $$\begin{gathered} S_{2}P−S_{1}P=c \\ \sqrt{{\left(x+a\right)}^2+y^2}−\sqrt{{\left(x-a\right)}^2+y^2}=c \end{gathered}$$

Squaring both of the sides of equation (1) , and you have

$$\begin{gathered} 2{(x+a)}^2+2y^2−c^2=2\sqrt{{(x+a)}^2+y^2}\sqrt{{(x−a)}^2+y^2} \\ {\left({\left(x+a\right)}^2+y^2−\frac{{\displaystyle c^2}}{{\displaystyle 2}}\right)}^2=\left({\left(x+a\right)}^2+y^2\right)\left({\left(x-a\right)}^2+y^2\right) \\ {(x+a)}^4+y^4+\frac{c^4}{4}+2y^2{(x+a)}^2−y^2{c}^2−c^2{(x+a)}^2={(x+a)}^2{(x−a)}^2+{(x+a)}^2{y}^2+{(x−a)}^2{y}^2+y^4 \end{gathered}$$

$$\begin{gathered} y^4+2y^2{(x+a)}^2−y^2{c}^2={(x+a)}^2{(x−a)}^2+{(x+a)}^2{y}^2+{(x−a)}^2{y}^2+y^4+{(x+a)}^4−\frac{c^4}{4}+c^2{(x+a)}^2 \\ 2y^2{(x+a)}^2−y^2{c}^2−{(x+a)}^2{y}^2−{(x−a)}^2{y}^2={(x+a)}^2{(x−a)}^2+{(x+a)}^4−\frac{c^4}{4}+c^2{(x+a)}^2 \end{gathered}$$

After solving the above equation gives:

$$\begin{gathered} y^2({(x+a)}^2−c^2−{(x−a)}^2)={(x+a)}^2({(x−a)}^2+{(x+a)}^2+c^2)−\frac{c^4}{4} \\ {y}^2(4xa−c^2)={(x+a)}^2(2x^2+2a^2+c^2)−\frac{c^4}{4} \\ {(x+a)}^2(2x^2+2a^2+c^2)−y^2(4xa−c^2)=\frac{c^4}{4} \end{gathered}$$

$\begin{array}{rcl}{(x+a)}^2×\frac{4(2x^2+2a^2+c^2)}{c^4}−y^2×\frac{4(4xa−c^2)}{c^4}& =& 1\\ \frac{x^2}{a^2}−\frac{y^2}{{\mathrm{β}}^2}& =& 1\end{array}$

Thus, the equation obtain is:

$\frac{x^2}{a^2}−\frac{y^2}{{\mathrm{β}}^2}=1$

Here, $\mathrm{β}$is a constant,

Above equation is an equation of hyperbola.

Hence, all curves are hyperbola.