Question

: How long is the minor axis for the ellipse shown below?

(x+4)^2 (y-1)^2
---------- + ---------- = 1
25 16

A: 8
B: 9
C: 12
D: 18

: Which of the following correctly represents the coordinates of the foci of the ellipse shown below?

(x-7)^2 (y+3)^2
--------- + ------------ = 1
4 16

A: (7, - 3 ± 2 √5)
B: (7 ± 2 √5, -3)
C: (7, - 3 ± 2 √3)
D: (7 ± 2 √3, - 3)

Answer 1

Answer on Question #57273 - Math - Analytic Geometry.

How long is the minor axis for the ellipse shown below?

$$\frac {(x + 4) ^ {2}}{25} + \frac {(y - 1) ^ {2}}{16} = 1.$$

Solution. For ellipse

$$\frac {(x - x _ {0}) ^ {2}}{a ^ {2}} + \frac {(y - y _ {0}) ^ {2}}{b ^ {2}} = 1$$

the minor axis is 2b. Then for ellipse

$$\frac {(x + 4) ^ {2}}{25} + \frac {(y - 1) ^ {2}}{16} = 1$$

the minor axis is $2 \cdot 4 = 8$ .

Answer: the minor axis is 8. (A)

Which of the following correctly represents the coordinates of the foci of the ellipse shown below?

$$\frac {(x - 7) ^ {2}}{4} + \frac {(y + 3) ^ {2}}{16} = 1.$$

Solution. Let the question of the ellipse

$$\frac {(x - x _ {0}) ^ {2}}{a ^ {2}} + \frac {(y - y _ {0}) ^ {2}}{b ^ {2}} = 1,$$

where $b \geq a$ . The coordinates of the foci of this ellipse are

$$(x _ {0}, y _ {0} \pm c),$$

where $c = \sqrt{b^2 - a^2}$

For the ellipse

$$\frac {(x - 7) ^ {2}}{4} + \frac {(y + 3) ^ {2}}{16} = 1$$ $$c = \sqrt {16 - 4} = \sqrt {12} = 2 \sqrt {3}.$$

The coordinates of the foci of this ellipse are

$$(7, - 3 \pm 2 \sqrt {3}).$$

Answer: the coordinates of the foci of the ellipse $(7, -3 \pm 2\sqrt{3})$ . (C)

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