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Precalculus : Hyperbolas and Ellipses

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Example Questions

Example Question #132 : Conic Sections

Find the endpoints of the major axis for the ellipse with the following equation:

$\frac{x^2}{64}+\frac{y^2}{49}=1$

Possible Answers:

(8,0)(0,8)

(0, 8)(0, -8)

(0,0)(0,8)

(8, 0)(-8, 0)

Correct answer:

(8, 0)(-8, 0)

Explanation:

Recall the standard form of the equation of an ellipse:

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

When a>b , the major axis is horizontal. In this case, (h-a, k) and (h+a, k) are the endpoints of the major axis.

When b>a , (h, k+b) and (h, k-b) are the endpoints of the major axis.

For the ellipse in question, (0,0) is the center. In addition, a=8 and b=7 . Since a>b , the major axis is horizontal and the endpoints are (8, 0) and (-8, 0)

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Example Question #41 : Hyperbolas And Ellipses

Find the endpoints of the major axis for the ellipse with the following equation:

$\frac{x^2}{100}+\frac{y^2}{400}=1$

Possible Answers:

(20, 0)(-20, 0)

(10, 0)(-10, 0)

(0, 20)(0, -20)

(10, 0)(20, 0)

Correct answer:

(0, 20)(0, -20)

Explanation:

Recall the standard form of the equation of an ellipse:

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

When a>b , the major axis is horizontal. In this case, (h-a, k) and (h+a, k) are the endpoints of the major axis.

When b>a , (h, k+b) and (h, k-b) are the endpoints of the major axis.

For the ellipse in question, (0,0) is the center. In addition, a=10 and b=20 . Since b>a , the major axis is vertical and the endpoints are (0, 20) and (0, -20) .

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Example Question #42 : Hyperbolas And Ellipses

Find the endpoints of the major axis of the ellipse with the following equation:

$\frac{(x-2)^2}{225}+\frac{(y+11)^2}{256}=1$

Possible Answers:

(17, 2)(-11, 13)

(17, -11)(-13, -11)

(-27, 2)(5, 2)

(2, -27)(2,5)

Correct answer:

(2, -27)(2,5)

Explanation:

Recall the standard form of the equation of an ellipse:

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

When a>b , the major axis is horizontal. In this case, (h-a, k) and (h+a, k) are the endpoints of the major axis.

When b>a , (h, k+b) and (h, k-b) are the endpoints of the major axis.

For the ellipse in question, (2,-11) is the center. In addition, a=15 and b=16 . Since b>a , the major axis is vertical and the endpoints are (2, -27) and (2, 5) .

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Example Question #44 : Hyperbolas And Ellipses

Find the endpoints of the major axis of the ellipse with the following equation:

$\frac{(x+5)^2}{169}+\frac{(y-3)^2}{144}=1$

Possible Answers:

(8, 3)(-18, 3)

(-5, 15)(3, -18)

(-5, 15)(-5, -9)

(3, 8)(-18, 3)

Correct answer:

(8, 3)(-18, 3)

Explanation:

Recall the standard form of the equation of an ellipse:

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

When a>b , the major axis is horizontal. In this case, (h-a, k) and (h+a, k) are the endpoints of the major axis.

When b>a , (h, k+b) and (h, k-b) are the endpoints of the major axis.

For the ellipse in question, (-5, 3) is the center. In addition, a=13 and b=12 . Since a>b , the major axis is horizontal and the endpoints are (8, 3) and (-18, 3)

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Example Question #45 : Hyperbolas And Ellipses

Find the endpoints of the major axis of the ellipse with the following equation:

25x^2+49y^2+200x-294y-284=0

Possible Answers:

(4, 8)(4, -2)

(3, 11)(3, -3)

(11, 3)(-3, 3)

(3, 11)(4, 8)

Correct answer:

(11, 3)(-3, 3)

Explanation:

Recall the standard form of the equation of an ellipse:

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

Start by putting the equation in the standard form as shown above.

Group the terms and terms together.

25x^2+49y^2+200x-294y-284=0

25x^2+200x+49y^2-294y-284=0

Factor out from the terms and from the terms.

25(x^2-8x)+49(y^2-6y)-284=0

Now, complete the squares. Remember to add the same amount to both sides of the equation!

25(x^2-8x+16)+49(y^2-6y+9)-284=841

Add 284 to both sides.

25(x^2-8x+16)+49(y^2-6y+9)=1125

Divide by 1125 on both sides.

$\frac{x^2-8x+16}{49}+\frac{y^2-6y+9}{25}=1$

Now factor both terms to get the standard form of the equation of an ellipse.

$\frac{(x-4)^2}{49}+\frac{(y-3)^2}{25}=1$

Recall that when a>b , the major axis is horizontal. In this case, (h-a, k) and (h+a, k) are the endpoints of the major axis.

When b>a , (h, k+b) and (h, k-b) are the endpoints of the major axis.

For the ellipse in question, (4, 3) is the center. In addition, a=7 and b=5 . Since a>b , the major axis is horizontal and the endpoints are (11, 3) and (-3, 3) .

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Example Question #46 : Hyperbolas And Ellipses

Find the endpoints of the major axis of the ellipse with the following equation:

9x^2+18y^2-108x-360y+1962=0

Possible Answers:

(6, 13)(6, 7)

$(6, 10+3\sqrt2)(6, 10-3\sqrt2)$

$(6+3\sqrt2, 10)(6-3\sqrt2, 10)$

$(3\sqrt2, 6)(-3\sqrt2, 6)$

Correct answer:

$(6+3\sqrt2, 10)(6-3\sqrt2, 10)$

Explanation:

Recall the standard form of the equation of an ellipse:

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

Start by putting the equation in the standard form as shown above.

Group the terms and terms together.

9x^2+18y^2-108x-360y+1962=0

9x^2-108x+18y^2-360y+1962=0

Factor out from the terms and from the terms.

9(x^2-12x)+18(y^2-20y)+1962=0

Now, complete the squares. Remember to add the same amount to both sides of the equation!

9(x^2-12x+36)+18(y^2-20y+100)+1962=2124

Subtract 1962 from both sides.

9(x^2-12x+36)+18(y^2-20y+100)=162

Divide by 162 on both sides.

$\frac{x^2-12x+36}{18}+\frac{y^2-20x+100}{9}=1$

Now factor both terms to get the standard form of the equation of an ellipse.

$\frac{(x-6)^2}{18}+\frac{(y-10)^2}{9}=1$

Recall that when a>b , the major axis is horizontal. In this case, (h-a, k) and (h+a, k) are the endpoints of the major axis.

When b>a , (h, k+b) and (h, k-b) are the endpoints of the major axis.

For the ellipse in question, (6, 10) is the center. In addition, $a=\sqrt{18}=3\sqrt2$ and b=3 . Since a>b , the major axis is horizontal and the endpoints are $(6+3\sqrt2, 10)$ and $(6-3\sqrt2, 10)$ .

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Example Question #42 : Understand Features Of Hyperbolas And Ellipses

Find the endpoints of the minor axis of the ellipse with the following equation:

$\frac{x^2}{100}+\frac{y^2}{169}=1$

Possible Answers:

(0, 13)(0, -13)

(10, 0)(-10, 0)

(0, 10)(0, -10)

(13, 10)(-13, 10)

Correct answer:

(10, 0)(-10, 0)

Explanation:

Recall the standard form of the equation of an ellipse:

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

When a<b , the minor axis is horizontal. In this case, (h-a, k) and (h+a, k) are the endpoints of the minor axis.

When b<a , (h, k+b) and (h, k-b) are the endpoints of the vertical minor axis.

For the ellipse in question, (0,0) is the center. In addition, a=10 and b=13 . Since a<b , the minor axis is horizontal and the endpoints are (10, 0) and (-10, 0)

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Example Question #48 : Hyperbolas And Ellipses

Find the endpoints of the minor axis of the ellipse with the following equation:

$\frac{x^2}{225}+\frac{y^2}{400}=1$

Possible Answers:

(15, 20)(-15, 20)

(0, 20)(0, -20)

(15, 0)(-15, 0)

(0, 15)(0, -15)

Correct answer:

(15, 0)(-15, 0)

Explanation:

Recall the standard form of the equation of an ellipse:

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

When a<b , the minor axis is horizontal. In this case, (h-a, k) and (h+a, k) are the endpoints of the minor axis.

When b<a , (h, k+b) and (h, k-b) are the endpoints of the vertical minor axis.

For the ellipse in question, (0,0) is the center. In addition, a=15 and b=20 . Since a<b , the minor axis is horizontal and the endpoints are (15, 0) and (-15, 0)

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Example Question #49 : Hyperbolas And Ellipses

Find the endpoints of the minor axis of the ellipse with the following equation:

$\frac{(x-5)^2}{25}+\frac{(y-1)^2}{36}=1$

Possible Answers:

(0, 1)(10, 1)

(1, 5)(10, 5)

(1, 0)(10, 1)

(5, 7)(5, -5)

Correct answer:

(0, 1)(10, 1)

Explanation:

Recall the standard form of the equation of an ellipse:

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

When a<b , the minor axis is horizontal. In this case, (h-a, k) and (h+a, k) are the endpoints of the minor axis.

When b<a , (h, k+b) and (h, k-b) are the endpoints of the vertical minor axis.

For the ellipse in question, (5,1) is the center. In addition, a=5 and b=6 . Since a<b , the minor axis is horizontal and the endpoints are (0,1) and (10, 1) .

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Example Question #50 : Hyperbolas And Ellipses

Find the endpoints of the minor axis of the ellipse with the following equation:

25x^2+20y^2+200x-120y+80=0

Possible Answers:

$(-4, 8)(-4, 2\sqrt5)$

(-4, 8)(-4, -2)

$(3, -4+2\sqrt5)(3, -4-2\sqrt5)$

$(-4+2\sqrt5, 3)(-4-2\sqrt5, 3)$

Correct answer:

$(-4+2\sqrt5, 3)(-4-2\sqrt5, 3)$

Explanation:

Recall the standard form of the equation of an ellipse:

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

Start by putting the equation in the standard form as shown above.

Group the terms and terms together.

25x^2+20y^2+200x-120y+80=0

25x^2+200x+20y^2-120y+80=0

Factor out from the terms and from the terms.

25(x^2+8x)+20(y^2-6y)+80=0

Now, complete the squares. Remember to add the same amount to both sides of the equation!

25(x^2+8x+16)+20(y^2-6y+9)+80=580

Subtract from both sides.

25(x^2+8x+16)+20(y^2-6y+9)=500

Divide by 500 on both sides.

$\frac{x^2+8x+16}{20}+\frac{y^2-6x+9}{25}=1$

Now factor both terms to get the standard form of the equation of an ellipse.

$\frac{(x+4)^2}{20}+\frac{(y-3)^2}{25}=1$

When a<b , the minor axis is horizontal. In this case, (h-a, k) and (h+a, k) are the endpoints of the minor axis.

When b<a , (h, k+b) and (h, k-b) are the endpoints of the vertical minor axis.

For the ellipse in question, (-4, 3) is the center. In addition, $a=\sqrt{20}=2\sqrt5$ and b=5 . Since a<b , the minor axis is horizontal and the endpoints are $(-4+2\sqrt5, 3)$ and $(-4-2\sqrt5, 3)$ .

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Precalculus : Hyperbolas and Ellipses

Study concepts, example questions & explanations for Precalculus

All Precalculus Resources

Example Questions

Example Question #132 : Conic Sections

Example Question #41 : Hyperbolas And Ellipses

Example Question #42 : Hyperbolas And Ellipses

Example Question #44 : Hyperbolas And Ellipses

Example Question #45 : Hyperbolas And Ellipses

Example Question #46 : Hyperbolas And Ellipses

Example Question #42 : Understand Features Of Hyperbolas And Ellipses

Example Question #48 : Hyperbolas And Ellipses

Example Question #49 : Hyperbolas And Ellipses

Example Question #50 : Hyperbolas And Ellipses

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