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Chapter 9: Q. 33 (page 772)

Use Cartesian coordinates to express the equations for the ellipses determined by the conditions specified in Exercises 32鈥�37.
$foci (卤 \sqrt{5}, 1), major axis 6$

Short Answer

Expert verified

The equation is $\frac{x^{2}}{9} + \frac{(y - 1)^{2}}{4} = 1 .$

Step by step solution

01

Step 1. Given information.

The given values are ,

$foci (卤 \sqrt{5}, 1), major axis 6$

02

Step 2. Value of variables.

The foci are $(\sqrt{5}, 1) (- \sqrt{5}, 1)$

$center = (\frac{\sqrt{5} - \sqrt{5}}{2}, \frac{1 + 1}{2}) = (0, 1)$

$G i v e n, 2 a = 6 a = 3 c = \sqrt{(0 - \sqrt{5})^{2} + (1 - 1)^{2}} [since D = \sqrt{{(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2}}] That is the distance from (\sqrt{5}, 1) (0, 1) . c = \sqrt{5} c^{2} = a^{2} - b^{2} (\sqrt{5})^{2} = 3^{2} - b^{2} b^{2} = 4$

03

Step 3. Substitution.

On substituting all the obtained values in,

$\frac{(x - h)^{2}}{a^{2}} + \frac{(y - k)^{2}}{b^{2}} = 1 where a > b, (h, k) is the center. \frac{(x - 0)^{2}}{9} + \frac{(y - 1)^{2}}{4} = 1 \frac{x^{2}}{9} + \frac{(y - 1)^{2}}{4} = 1$

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