Mathematics, 22.04.2020 02:38 sha712

Eliminate the parameter and obtain the standard form of the rectangular equation. Ellipse: x = h + a cos(θ), y = k + b sin(θ) Use your result to find a set of parametric equations for the line or conic. (When 0 ≤ θ ≤ 2π. Enter your answers as a comma-separated list of equations.) Ellipse: vertices: (−4, 0), (6, 0); foci: (−2, 0), (4, 0)

Answers: 1

Show answers

Answers

Answer from: mamshie

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

b) $x=1+5 \,cos(\theta)\,,\,y=4\,sin(\theta)$

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

Step-by-step explanation:

Start by isolating the trigonometric expression in both equations, ad then use the Pythagorean identity:

$cos^2(\theta)+sin^2(\theta)=1$

to obtain the standard equation of a conic.

$x=h+a\,cos(\theta)\\x-h=a\,cos(\theta)\\cos(\theta)=\frac{x-h}{a}$ $y=k+b\,sin(\theta)\\y-k=b\,sin(\theta)\\sin(\theta)=\frac{y-k}{b}$

then:

$cos^2(\theta)+sin^2(\theta)=1\\(\frac{x-h}{a})^2+ (\frac{y-k}{b})^2=1$

this is the equation of an ellipse centered at (h,k), and with horizontal axis length = 2a , and vertical axis length = 2b.

The parametric equations are those we started with:

$x=h+a \,cos(\theta)\,,\,y=k+b\,sin(\theta)$ but we need to find the appropriate parameters for the requested ellipse, as shown below.

For an ellipse of vertices (-4,0) a,d (6,0) and foci at (-2,0) and (4.0), we are dealing with an ellipse with major horizontal axis on the line y=0, and major diameter length of 10 units, so the parameter a=5 . The center of the ellipse is therefore at (1,0).

We recall that the vertices of a translated horizontal ellipse are located at (a+h,k) and (-a+h,k) , then k=0 to satisfy the information given (-4,0) & (6,0), and since a=5 , we deduce that a+h = 6 and therefore h=1.

To find "b" (the only parameter missing for the standard equation of the conic), we need the information on the foci (-2,0) and (4,0) which must equal (h-c,k) and (h+c,k) with k=0 and h=1 which then gives that c=3

Now using the formula for the parameter "c" of the foci: $c=\sqrt{a^2-b^2}$

$c=\sqrt{a^2-b^2}\\3=\sqrt{5^2-b^2}\\9=25-b^2\\b^2=25-9\\b^2=16\\b=4$

Then we can write the equation of this ellipse in standard form as:

$(\frac{x-h}{a})^2+ (\frac{y-k}{b})^2=1\\(\frac{x-1}{5})^2+ (\frac{y-0}{4})^2=1\\\\(\frac{x-1}{5})^2+ (\frac{y}{4})^2=1\\\frac{(x-1)^2}{25}+ \frac{y^2}{16}=1$

Answer from: Quest

n=9

step-by-step explanation:

7 times a number = 7n

decreased by 13 = 7n-13

same as is =

5 times the number = 5n

increased by 5 = 5n+5

7n-13 = 5n+5

2n=18

n=9

Answer from: Quest

the option best identifies the textual evidence that supports this conclusion is c.) when an argument is over, how many weighty reasons does a man recollect which his heat and violence made him utterly forget? in order to recognize textual evidence is to identify the topic sentence where the author states the main idea. textual evidence is what supports the topic sentence or main idea of a passage.

Another question on Mathematics

Mathematics, 21.06.2019 17:20

The function a represents the cost of manufacturing product a, in hundreds of dollars, and the function b represents the cost of manufacturing product b, in hundreds of dollars. a(t) = 5t + 2 b(t) = 7t - 2t +4 find the expression that describes the total cost of manufacturing both products, a(t) + b(t). a. 7t^2 - 7t + 2 b. 7t^2 + 3 +6 c. 7t^2 + 7t - 6 d. 7t^2 - 3t + 6

Answers: 3

Answer

Mathematics, 21.06.2019 20:10

60 2(x + 2) in the diagram, gef and hef are congruent. what is the value of x?

Answers: 2

Answer

Mathematics, 21.06.2019 21:10

For each sentence below, find the value of x that makes each sentence true. (515) 5 = 25* x= (813)2 = 4* x=

Answers: 2

Answer

Mathematics, 21.06.2019 23:30

What is the measure of angle x? a.101 b.158 c.44 d.79