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Precalculus : Pre-Calculus

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Example Questions

Example Question #1 : Understand Features Of Hyperbolas And Ellipses

Find the equation of an ellipse centered at the origin if the major axis is parallel to the x-axis and has a length of units and the minor axis has a length of units.

Possible Answers:

$\small \frac{x^2}{16}+\frac{y^2}{4}=1$

$\small \frac{x^2}{64}+\frac{y^2}{16}=1$

$\small \small \frac{x^2}{16}+\frac{y^2}{64}=1$

$\small \small \frac{x^2}{4}+\frac{y^2}{16}=1$

Correct answer:

$\small \frac{x^2}{16}+\frac{y^2}{4}=1$

Explanation:

The formula for an ellipse centered at the point (h, k) with a horizontal major axis (ie: parallel to the x-axis) has the formula

$\small \frac{(x-h)^2}{[\frac{a}{2}]^2}+\frac{(y-k)^2}{[\frac{b}{2}]^2}=1$

where $\small a$ and $\small b$ are the lengths of the major and minor axes respectively.

Since the origin is at $\small (0,0)$ and $\small a=8$ and $\small b=4$ in this problem, the equation is

$\small \frac{(x-0)^2}{[\frac{8}{2}]^2}+\frac{(y-0)^2}{[\frac{4}{2}]^2}=1$

$\small \frac{x^2}{16}+\frac{y^2}{4}=1$

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Example Question #1 : Understand Features Of Hyperbolas And Ellipses

The equation of an ellipse, , is $\frac{(x+3)^2}{9}+\frac{(y-7)^2}{36}=1$ . Which of the following is the correct center and foci of this ellipse?

Possible Answers:

Center= (3,-7) , Foci= (-2,7) and (8,7)

Center= (-3,7) , Foci= (-8,7) and (2,7)

Center= (-3,7) , Foci= (-3,2) and (-3,12)

Center= (3,-7) , Foci= (3,-2) and

Center= (3,6) , Foci= (3,1) and (3,11)

Correct answer:

Center= (-3,7) , Foci= (-3,2) and (-3,12)

Explanation:

Because our equation is already in the format

$\frac{(x-h)^2}{w^2}+\frac{(y-k)^2}{v^2}=1$ ,

we do not have to manipulate the equation. The center of any ellipse in this form will always be . So in this case, our center will be (-3,7) . To find the foci of the ellipse, we must use the equation b^2+c^2=a^2 , where a^2 is the greater of the two denominators in our equation ( w^2 and v^2 ), b^2 is the lesser and is the distance from the center to the foci.

We know that a^2=36 and b^2=9 .

By using a^2-b^2=c^2 , we see that c^2=25 , so c=5 .

We now know that the two foci are going to be units in either direction of the center along the greater axis.

Because the greater denominator is under our term containing , our ellipse will have its greater axis going vertically, rather than horizontally. Therefore, our foci will be units above and below our center, at (-3,2) and (-3,12) .

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Example Question #4 : Hyperbolas And Ellipses

Find the center of the ellipse with the following equation:

$\frac{(x-1)^2}{36}+\frac{(y+2)^2}{9}=1$

Possible Answers:

(-1, 2)

(1, -2)

(-6, -3)

(6, 3)

Correct answer:

(1, -2)

Explanation:

Recall that the standard form of the equation of an ellipse is

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

For the equation given in the question, h=1 and k=-2 .

The center of the ellipse is at (1, -2)

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Example Question #4 : Understand Features Of Hyperbolas And Ellipses

Find the center of the ellipse with the following equation:

$\frac{(x+9)^2}{2}+\frac{(y-4)^2}{3}=1$

Possible Answers:

$(\sqrt2, \sqrt3)$

(9, -4)

$(-\sqrt2, -\sqrt3)$

(-9, 4)

Correct answer:

(-9, 4)

Explanation:

Recall that the standard form of the equation of an ellipse is

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

For the equation given in the question, h=-9 and k=4 .

The center of the ellipse is at (-9, 4) .

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Example Question #11 : Hyperbolas And Ellipses

Find the center of the ellipse with the following equation:

$\frac{(x+4)^2}{2}+\frac{(y-2)^2}{8}=1$

Possible Answers:

$(-\sqrt2, -2\sqrt2)$

$(\sqrt2, 2\sqrt2)$

(4, -2)

(-4, 2)

Correct answer:

(-4, 2)

Explanation:

Recall that the standard form of the equation of an ellipse is

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

For the equation given in the question, h=-4 and k=2 .

The center of the ellipse is at (-4, 2) .

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Example Question #11 : Hyperbolas And Ellipses

Find the center of the ellipse with the following equation:

25x^2+9y^2+200x-36y+211=0

Possible Answers:

(-4, 2)

(3, 5)

(25, 9)

(2, 4)

Correct answer:

(-4, 2)

Explanation:

Start by putting the equation back into the standard equation of the ellipse:

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

Group the terms and terms together.

25x^2+9y^2+200x-36y+211=0

25x^2+200x+9y^2-36y+211=0

Factor out a from the terms, and a from the terms.

25(x^2+8x)+9(y^2-4y)+211=0

Now, complete the square. Remember to add the same amounts on both sides of the equation.

25(x^2+8x+16)+9(y^2-4y+4)+211=436

25(x^2+8x+16)+9(y^2-4y+4)=225

Now, divide both sides by 225 .

$\frac{x^2+8x+16}{9}+\frac{y^2-4y+4}{25}=1$

Finally, factor the equations to get the standard form of the equation for an ellipse.

$\frac{(x+4)^2}{9}+\frac{(y-2)^2}{25}=1$

Since h=-4 and k=2 , the center for this ellipse is (-4, 2) .

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Example Question #12 : Hyperbolas And Ellipses

Find the center of the ellipse with the following equation:

4x^2+6y^2-40x-36y+130=0

Possible Answers:

(-3, 5)

(40, 36)

(4, 6)

(5, 3)

Correct answer:

(5, 3)

Explanation:

Start by putting the equation back into the standard equation of the ellipse:

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

Group the terms and terms together.

4x^2+6y^2-40x-36y+130=0

4x^2-40x+6y^2-36y+130=0

Factor out a from the terms and a from the terms.

4(x^2-10x)+6(y^2-6y)+130=0

Now, complete the squares. Make sure you add the same amount on both sides!

4(x^2-10x+25)+6(y^2-6y+9)+130=154

Subtract 130 from both sides.

4(x^2-10x+25)+6(y^2-6y+9)=24

Now, divide both sides by .

$\frac{x^2-10x+25}{6}+\frac{y^2-6y+9}{4}=1$

Finally, factor the terms to get the standard form of the equation of an ellipse.

$\frac{(x-5)^2}{6}+\frac{(y-3)^2}{4}=1$

Since h=5 and k=3 , the center of the ellipse is (5, 3) .

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Example Question #11 : Understand Features Of Hyperbolas And Ellipses

Find the center of the ellipse with the following equation:

9x^2+8y^2-18x-112y+329=0

Possible Answers:

(6, 12)

(9, 8)

(1, 7)

(-1, -7)

Correct answer:

(1, 7)

Explanation:

Start by putting the equation back into the standard equation of the ellipse:

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

Group the terms and terms together.

9x^2+8y^2-18x-112y+329=0

9x^2-18x+8y^2-112y+329=0

Factor out a from the terms and a from the terms.

9(x^2-2x)+8(y^2-14y)+329=0

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

9(x^2-2x+1)+8(y^2-14y+49)+329=401

Subtract 329 from both sides.

9(x^2-2x+1)+8(y^2-14y+49)=72

Divide both sides by .

$\frac{x^2-2x+1}{8}+\frac{y^2-14y+49}{9}=1$

Finally, factor the terms to get the standard form of the equation of an ellipse.

$\frac{(x-1)^2}{8}+\frac{(y-7)^2}{9}=1$

Since h=1 and k=7 , (1, 7) is the center of this ellipse.

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Example Question #15 : Understand Features Of Hyperbolas And Ellipses

Find the foci of an ellipse with the following equation:

$\frac{(x-5)^2}{25}+\frac{(y-1)^2}{169}=1$

Possible Answers:

(169, 25), (25, 169)

(5, 13), (5, -11)

(-5, -13), (-5, 11)

(17, 1), (-7, 1)

Correct answer:

(5, 13), (5, -11)

Explanation:

Recall that the standard form of the equation of an ellipse is

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

When a^2>b^2 , the major axis will lie on the -axis and be horizontal. When b^2>a^2 , the major axis will lie on the -axis and be vertical.

Recall also that the distance from the center to a focus, , is given by the equation $c=\sqrt{a^2-b^2}$ when a>b , and the equation is $c=\sqrt{b^2-a^2}$ when b>a .

When the major axis follows the -axis, the points for the foci are (h+c, k) and (h-c, k) .

When the major axis follows the -axis, the points for the foci are (h, k+c) and (h, k-c) .

For the given equation, the center is at (5, 1) . Since b^2>a^2 , the major-axis is vertical.

Plug in the values to solve for .

$c=\sqrt{169-25}=\sqrt{144}=12$

Now, add to the y-coordinate of the center to get one focus. Subtract from the y-coordinate of the center to get the other focus point.

The foci for the ellipse is then (5, 13) and (5, -11) .

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Example Question #106 : Conic Sections

Find the foci of the ellipse with the following equation:

5x^2+6y^2+50x+72y+311=0

Possible Answers:

(-5, -4), (-5, -7)

(-5, 2), (-5, 1)

(6, 6)(7, 6)

(-6, -6), (-4, -6)

Correct answer:

(-6, -6), (-4, -6)

Explanation:

Recall that the standard form of the equation of an ellipse is

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

When a^2>b^2 , the major axis will lie on the -axis and be horizontal. When b^2>a^2 , the major axis will lie on the -axis and be vertical.

Recall also that the distance from the center to a focus, , is given by the equation $c=\sqrt{a^2-b^2}$ when a>b , and the equation is $c=\sqrt{b^2-a^2}$ when b>a .

When the major axis follows the -axis, the points for the foci are (h+c, k) and (h-c, k) .

When the major axis follows the -axis, the points for the foci are (h, k+c) and (h, k-c) .

Start by putting the equation into the standard form of the equation of an ellipse.

Group the and terms together.

5x^2+6y^2+50x+72y+311=0

5x^2+50x+6y^2+72y+311=0

Now, factor out a from the terms and a from the terms.

5(x^2+10x)+6(y^2+12y)+311=0

Complete the squares. Remember to add the same amount to both sides of the equation!

5(x^2+10x+25)+6(y^2+12y+36)+311=341

Subtract 311 from both sides of the equation.

5(x^2+10x+25)+6(y^2+12y+36)=30

Divide both sides by .

$\frac{x^2+10x+25}{6}+\frac{y^2+12y+36}{5}=1$

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

$\frac{(x+5)^2}{6}+\frac{(y+6)^2}{5}=1$

Now, the center for this ellipse is (-5, -6) and its major axis is horizontal.

Next, solve for .

$c=\sqrt{6-5}=\sqrt1=1$

The foci for this ellipse are then at (-6, -6) and (-4, -6) .

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Precalculus : Pre-Calculus

Study concepts, example questions & explanations for Precalculus

All Precalculus Resources

Example Questions

Example Question #1 : Understand Features Of Hyperbolas And Ellipses

Example Question #1 : Understand Features Of Hyperbolas And Ellipses

Example Question #4 : Hyperbolas And Ellipses

Example Question #4 : Understand Features Of Hyperbolas And Ellipses

Example Question #11 : Hyperbolas And Ellipses

Example Question #11 : Hyperbolas And Ellipses

Example Question #12 : Hyperbolas And Ellipses

Example Question #11 : Understand Features Of Hyperbolas And Ellipses

Example Question #15 : Understand Features Of Hyperbolas And Ellipses

Example Question #106 : Conic Sections

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