Q. 43 Use Cartesian coordinates to exp... [FREE SOLUTION]

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Chapter 9: Q. 43 (page 772)

Use Cartesian coordinates to express the equations for the hyperbolas determined by the conditions specified in Exercises 38鈥�43.
$foci (3, 1) and (3, 9), directrix y = 4$

Short Answer

Expert verified

The equation is $\frac{(y - 5)^{2}}{4} - \frac{(x - 3)^{2}}{12} = 1$ .

Step by step solution

Step 1. Given information.

It is given that $foci (3, 1) and (3, 9), directrix y = 4$

Step 2. Value of the variables.

Now,

$The focus points are (3, 1) (3, 9) . Center = (\frac{3 + 3}{2}, \frac{1 + 9}{2}) [since mid point = (\frac{x_{1} + x_{2}}{2}, \frac{y_{1} + y_{2}}{2})] Center = (3, 5) Given driectries are y = 4 . That means \frac{b}{e} = 4 \frac{b}{4} = e Then b e = 1 b^{2} = 4 c = \sqrt{(3 - 3)^{2} + (5 - 1)^{2}} [since D = \sqrt{{(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2}}] c = 4 For a hyperbola, a^{2} + b^{2} = c^{2} a^{2} + 4 = 4^{2} a^{2} = 12$

Step 3. Substitution.

Now, substitute the obtained values,

$\frac{(y - k)^{2}}{b^{2}} - \frac{(x - h)^{2}}{a^{2}} = 1 where (h, k) is the center. \frac{(y - 5)^{2}}{4} - \frac{(x - 3)^{2}}{12} = 1 [since a^{2} = 12, b^{2} = 4] \frac{(y - 5)^{2}}{4} - \frac{(x - 3)^{2}}{12} = 1$

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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.
$foci (3, 1) and (3, 9), directrix y = 4$

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.

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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(3,1)and(3,9), directrixy=4

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

Applied Mathematics

Decision Maths

Theoretical and Mathematical Physics

Statistics

Mechanics Maths

Geometry

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.
$foci (3, 1) and (3, 9), directrix y = 4$