Chapter 9: Q. 37 (page 772)

Use Cartesian coordinates to express the equations for the ellipses determined by the conditions specified in Exercises 32–37.
$foci (0, \pm α), minor axis 2 α$

Short Answer

Expert verified

The answer is $\frac{x^{2}}{α^{2}} + \frac{y^{2}}{2 α^{2}} = 1 .$

Step by step solution

Step 1. Given information.

The given values are,

$foci (0, \pm α), minor axis 2 α$ .

Step 2. values of the variables.

The foci are $(0, α) (0, - α)$

Centre $= (\frac{0 + 0}{2}, \frac{α - α}{2}) [since mid point = (\frac{x_{1} + x_{2}}{2}, \frac{y_{1} + y_{2}}{2})]$ $= (0, 0)$

$Given minor axis is 2 α . Then 2 b = 2 α b = α c = \sqrt{(0 - 0)^{2} + (0 - α)^{2}} since D = \sqrt{{(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2}}] That is the distance from (0, α) (0, 0) . c = α Now take c^{2} = a^{2} - b^{2} c^{2} = a^{2} - b^{2} α^{2} = a^{2} - α^{2} a^{2} = 2 α^{2}$

Step 3. Substitution.

On substituting,

$\frac{(x - h)^{2}}{b^{2}} + \frac{(y - k)^{2}}{a^{2}} = 1 where a > b, (h, k) Is the center. \frac{(x - 0)^{2}}{α^{2}} + \frac{(y - 0)^{2}}{2 α^{2}} = 1 \frac{x^{2}}{α^{2}} + \frac{y^{2}}{2 α^{2}} = 1 .$

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Use Cartesian coordinates to express the equations for the ellipses determined by the conditions specified in Exercises 32–37.
$foci (0, \pm α), minor axis 2 α$

Short Answer

Step by step solution

Step 1. Given information.

Step 2. values of the variables.

Step 3. Substitution.

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Use Cartesian coordinates to express the equations for the ellipses determined by the conditions specified in Exercises 32–37.foci(0,±α), minor axis2α

Short Answer

Step by step solution

Step 1. Given information.

Step 2. values of the variables.

Step 3. Substitution.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Probability and Statistics

Geometry

Statistics

Calculus

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

Use Cartesian coordinates to express the equations for the ellipses determined by the conditions specified in Exercises 32–37.
$foci (0, \pm α), minor axis 2 α$