Chapter 9: Q. 40 (page 725)

Use Cartesian coordinates to express the equations for the hyperbolas determined by the conditions specified in Exercises 38–43.
$foci (0, \pm 4), directrices y = \pm 1$

Short Answer

Expert verified

The equation is $\frac{y^{2}}{4} - \frac{x^{2}}{12} = 1$ .

Step by step solution

Step 1. Given information.

We are given,

$foci (0, \pm 4), directrices y = \pm 1$

Step 2. Value of the variables.

Now, as given,

$The focus points are (0, 4), (0, - 4) . Center = (\frac{0 + 0}{2}, \frac{4 - 4}{2}) [since mid point = (\frac{x_{1} + x_{2}}{2}, \frac{y_{1} + y_{2}}{2})] Center = (0, 0) Given directries are y = \pm 1 . That means \frac{b}{e} = 1 b = e Then b e = 4 b \cdot b = 4 [since e = b] b^{2} = 4 c = \sqrt{(0 - 4)^{2} + (0 - 0)^{2}} [since D = \sqrt{{(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2}}] That is the distance from (0, 4) (0, 0) c = 4 For a hyperbola, a^{2} + b^{2} = c^{2} a^{2} + 4 = 4^{2} a^{2} = 12$

Step 3. Substitution.

Now, substitute the obtained values,

$\frac{(y - k)^{2}}{b^{2}} - \frac{(x - h)^{2}}{a^{2}} = 1 where (h, k) is the center. \frac{(y - 0)^{2}}{4} - \frac{(x - 0)^{2}}{12} = 1 [since a^{2} = 12, b^{2} = 4] \frac{y^{2}}{4} - \frac{x^{2}}{12} = 1$

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Use Cartesian coordinates to express the equations for the hyperbolas determined by the conditions specified in Exercises 38–43.
$foci (0, \pm 4), directrices y = \pm 1$

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(0,±4), directricesy=±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

Pure Maths

Geometry

Statistics

Decision Maths

Probability and Statistics

Mechanics Maths

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.
$foci (0, \pm 4), directrices y = \pm 1$