Chapter 9: Q. 38 (page 772)

Use Cartesian coordinates to express the equations for the hyperbolas determined by the conditions specified in Exercises 38–43.
$foci (\pm 2, 0), directrices x = \pm 1$

Short Answer

Expert verified

The equation is $x^{2} - y^{2} = 2 .$

Step by step solution

Step 1. Given Information.

The given values are,

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

Step 2. Value of the Variables.

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

$Center = (\frac{2 - 2}{2}, \frac{0 + 0}{2}) [since mid point = (\frac{x_{1} + x_{2}}{2}, \frac{y_{1} + y_{2}}{2})] Center = (0, 0) Given driectries are x = \pm 1 . That means \frac{a}{e} = 1 a = e Then a e = 2 a^{2} = 2 c = \sqrt{(0 - 2)^{2} + (0 - 0)^{2}} [since D = \sqrt{{(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2}}] c = 2 For a hyperbola, a^{2} + b^{2} = c^{2} 2 + b^{2} = 2^{2} b^{2} = 2$

Step 3. Substitution.

Substituting the given values,

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

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Use Cartesian coordinates to express the equations for the hyperbolas determined by the conditions specified in Exercises 38–43.
$foci (\pm 2, 0), directrices x = \pm 1$

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Value of the Variables.

Step 3. Substitution.

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Use Cartesian coordinates to express the equations for the hyperbolas determined by the conditions specified in Exercises 38–43.foci(±2,0), directricesx=±1

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Value of the Variables.

Step 3. Substitution.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Statistics

Probability and Statistics

Theoretical and Mathematical Physics

Decision Maths

Logic and Functions

Study anywhere. Anytime. Across all devices.

Use Cartesian coordinates to express the equations for the hyperbolas determined by the conditions specified in Exercises 38–43.
$foci (\pm 2, 0), directrices x = \pm 1$