Q. 40 Use Cartesian coordinates to exp... [FREE SOLUTION]

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Chapter 9: Q. 40 (page 725)

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

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,

$foci (0, 卤 4), directrices y = 卤 1$

Step 2. Value of the variables.

Now, as given,

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

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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91影视

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

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Value of the variables.

Step 3. Substitution.

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91影视

Use Cartesian coordinates to express the equations for the hyperbolas determined by the conditions specified in Exercises 38鈥�43.foci(0,卤4), directricesy=卤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

Calculus

Probability and Statistics

Applied Mathematics

Decision Maths

Statistics

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鈥�43.
$foci (0, 卤 4), directrices y = 卤 1$