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Mice21 [21]
2 years ago
15

Which is the equation of an ellipse with directrices at x = ±4 and foci at (2, 0) and (−2, 0)?

Mathematics
1 answer:
NemiM [27]2 years ago
6 0

Answer:

x squared over 4 plus y squared over 8 equals 1

Step-by-step explanation:

The general equation of ellipse is given as;

(x²/a²) + (y²/b²) = 1

The coordinates of a foci are: (±c, 0) where;

c² = b² - a²

However, we know that equation of directrix is; x = ±a/e

Now, Directrix is given ±4

Thus, a/e = 4

a = 4e

We also know that c = ae from ellipse foci coordinates.

Thus, ae = 2

since ae = 2, then (4e)e = 2

4e² = 2

e² = 2/4

e = 1/2

Thus;

a = 4 × 1/2

a = 2

Since c² = b² - a²;

2² = b² - 2²

4 = b² - 4

b² = 8

From (x²/a²) + (y²/b²) = 1, we can put our values to get;

x²/4 + y²/8 = 1

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Step-by-step explanation:

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D. The question is not referring to a standard deviation

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2 years ago
Are these fractions equivalent or nonequivalent?<br> 2/9 4/18
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Answer: they are equivalent

Step-by-step explanation:

6 0
3 years ago
Show that: (sec theta - cosec theta) (1 + tan theta + cot theta) = sec theta tan theta - cosec theta cot theta)
kvv77 [185]

Answer:

see derivation below

Step-by-step explanation:

Show that:

( sec(t) - cosec(t) ) ( 1 + tan(t) + cot(t) ) =

sec(t) tan(t) - cosec(t) cot(t)

Some trigonometric definitions used:

tan(t) = sin(t)/cos(t)

cot(t) = cos(t)/sin(t)

sec(t) = 1/cos(t)

csc(t) = 1/sin(t)

some trigonometric identities used:

sin^2(t) + cos^2(t) = 1 ......................(1)

rewrite left-hand side in terms of sine and cosine

(1/cos(t) - 1/sin(t) ) ( 1 + sin(t)/cos(t) + cos(t)/sin(t) )

Simplify using common denominator sin(t)cos(t)

= ( (sin(t) - cos(t))/(sin(t)*cos(t)) ) * ( ( sin(t)cos(t) + sin^2(t) + cos^2(t)) / ( sin(t)cos(t) ) )

= ( sin(t) -cos(t) ) * (1 + sin(t)cos(t) ) / ( sin^2(t) cos^2(t) )   ...... using (1)

Expand by multiplication

= ( sin(t) -cos(t) + sin^2(t)cos(t) - sin(t)cos^2(t) ) / ( sin^2(t) cos^2(t) )

Rearrange by factoring out sin(t) and cos(t) in numerator

= ( sin(t) (1-cos^2(t) - cos(t)(1-sin^2(t) )  / ( sin^2(t) cos^2(t) )

= ( sin^3(t) - cos^3(t) ) /( sin^2(t) cos^2(t) )   .........................using (1)

Cancel common factors

= sin(t)/(cos^2(t)) - cos(t)/(sin^2(t))

Rewrite using trigonometric definitions

= sec(t)tan(t) - csc(t)cot(t)   as in Right-Hand Side

4 0
3 years ago
At a restaurant, four people ordered fried crab cakes and four people ordered a cup of gumbo, with a total of $31. If only two p
ruslelena [56]

Answer:

Fried crab cakes = $4.50

Cup of gumbo = $3.25

Step-by-step explanation:

Let's make the price of fried crab cakes = x and the price of a cup of gumbo = y. Set up two equations for x and y and solve for each variable.

<h3><u>Setting up the equations:</u></h3>

4 people ordered x and 4 people ordered y for a total of 31 dollars, so this equation would look like: 4x + 4y = 31

2 people ordered x and 1 person ordered y for a total of $12.25, so this equation would look like: 2x + y = 12.25

<h3><u>Substitution method:</u></h3>

Now you have your two equations-- solve for a variable in one of the equations and substitute that value into the other equation.

The easiest way to start would be to solve for y in the second equation by subtracting 2x from both sides.

y = 12.25 - 2x

Substitute this value into the first equation.

4x + 4(12.25 - 2x) = 31

Now solve for x-- start by distributing 4 inside the parentheses.

4x + (49 - 8x) = 31

Combine like terms.

-4x + 49 = 31

Subtract 49 from both sides.

-4x = -18

Divide both sides by -4.

x = 4.5

Substitute this value of x into our starting equation for which we solved for y.

2(4.5) + y = 12.25

Multiply 2 and 4.5 together.

9 + y = 12.25

Subtract 9 from both sides.

y = 3.25

Each order of fried crab cakes is worth the x-value [$4.50] and each cup of gumbo is worth the y-value [$3.25].

7 0
3 years ago
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