Q. 38 - Calculus

Question

Use Cartesian coordinates to express the equations for the hyperbolas determined by the conditions specified in Exercises 38–43. 

$\text{ foci }(\pm 2,0)\text{, directrices }x=\pm 1$

Step-by-Step Solution

Answer

The equation is $x^2-y^2=2.$

Step 1. Given Information.

The given values are,

$\text{ foci }(\pm 2,0)\text{, directrices }x=\pm 1$

Step 2. Value of the Variables.

The foci are $(2,0)(-2,0).$

$$\begin{gathered} \text{ Center }=\left(\frac{2-2}{2},\frac{0+0}{2}\right)\left[\text{ since mid point }=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)\right] \\ \text{ Center }=(0,0) \\ \text{ Given driectries are }x=\pm 1\text{. } \\ \text{ That means }\frac{a}{e}=1 \\ a=e \\ \text{ Then }ae=2 \\ {a}^2=2 \\ c=\sqrt{(0-2{)}^2+(0-0{)}^2}\left[\text{ since }D=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}\right] \\ c=2 \\ \text{ For a hyperbola, }{a}^2+b^2=c^2 \\ 2+b^2=2^2 \\ {b}^2=2 \end{gathered}$$

Step 3. Substitution.

Substituting the given values,

$$\begin{gathered} \frac{(x-h{)}^2}{a^2}-\frac{(y-k{)}^2}{b^2}=1\text{ where }(h,k)\text{ is the center. } \\ \frac{(x-0{)}^2}{2}-\frac{(y-0{)}^2}{2}=1 \\ \frac{x^2}{2}-\frac{y^2}{2}=1 \left[\text{ since }{a}^2=2,b^2=2\right] \\ {x}^2-y^2=2. \end{gathered}$$