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Alisiya [41]
3 years ago
6

What is the rectangular equivalence to the parametric equations?

Mathematics
1 answer:
denpristay [2]3 years ago
8 0

Answer:

For x(θ) = 3cosθ + 2

y² = [9x²/(x - 2)²] - x²

For x(θ) = 3cosθ + 2

x² = [4y²/(y - 1)²] - y²

Step-by-step explanation:

Given the following equivalence:

x² + y² = r²

r = √(x² + y²)

x = rcosθ

cosθ = x/r

y = rsinθ

sinθ = y/r

Applying these to the given equations,

x(θ) = 3cosθ + 2

x = 3(x/r) + 2

xr = 3x + 2r

(x - 2)r = 3x

r = 3x/(x - 2)

Square both sides

r² = 9x²/(x - 2)²

(x - 2)²r² = 9x²

(x - 2)²(x² + y²) = 9x²

(x² + y²) = 9x²/(x - 2)²

y² = [9x²/(x - 2)²] - x²

y(θ) = 2sinθ - 1

y = 2y/r - 1

yr = 2y - r

(y - 1)r = 2y

r = 2y/(y - 1)

Square both sides

r² = 4y²/(y - 1)²

x² + y² = 4y²/(y - 1)²

x² = [4y²/(y - 1)²] - y²

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The endpoints of one diagonal of a rhombus are (-7, -2) and (-1, -4). If the coordinates of the 3rd vertex are (-6, -9), what ar
podryga [215]

The coordinates of the 4th vertex of the rhombus are (-2,3)

<h3>How to determine the coordinates of the 4th vertex?</h3>

The diagonals are given as:

(-7, -2) and (-1, -4)

The 3rd vertex is given as: (-6,-9)

Calculate the distance between the vertices of the diagonals and the 3rd vertex using:

d = \sqrt{(x_2 -x_1)^2 + (y_2 - y_1)^2}

So, we have:

d_1 = \sqrt{(-7 + 6)^2 + (-2 + 9)^2} =\sqrt{50

d_2 = \sqrt{(-1 + 6)^2 + (-4 + 9)^2} =\sqrt{50

Let the 4th vertex be (x,y)

So, we have:

d_3 = \sqrt{(-7 - x)^2 + (-2 - y)^2} =\sqrt{50

d_4 = \sqrt{(-1 - x)^2 + (-4 - y)^2} =\sqrt{50

Equate d3 and d4

\sqrt{(-1 - x)^2 + (-4 - y)^2} = \sqrt{(-7 - x)^2 + (-2 - y)^2}

Take the square of both sides

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

Expand

1 + 2x + x^2 + 16 + 8y + y^2 = 49 + 14x + x^2 + 4 + 4y + y^2

Evaluate the like terms

1 + 2x  + 16 + 8y  = 49 + 14x + 4 + 4y

Collect like terms

14x - 2x + 4y - 8y = 1 + 16 - 49 - 4

12x - 4y = -36

Divide through by 4

3x - y = -9

Next, we test the options in the above equation

Point (-8,13) means x = -8 and y = 13

So, we have:

3x - y = -9

3(-8) - 13 = -9

-37 = -9 --- this is false

Point (-2, 3) means x = -2 and y = 3

So, we have:

3x - y = -9

3(-2) - 3 = -9

-9 = -9 --- this is true.

Hence, the coordinates of the 4th vertex are (-2,3)

Read more about rhombus at:

brainly.com/question/20627264

#SPJ1

8 0
1 year ago
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