These are two questions and two answers.
Question 1) Which of the following polar equations is equivalent to the parametric equations below?
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x=t²
y=2t</span>
Answer: option <span>A.) r = 4cot(theta)csc(theta)
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Explanation:
1) Polar coordinates ⇒ x = r cosθ and y = r sinθ
2) replace x and y in the parametric equations:
r cosθ = t²
r sinθ = 2t
3) work r sinθ = 2t
r sinθ/2 = t
(r sinθ / 2)² = t²
4) equal both expressions for t²
r cos θ = (r sin θ / 2 )²
5) simplify
r cos θ = r² (sin θ)² / 4
4 = r (sinθ)² / cos θ
r = 4 cosθ / (sinθ)²
r = 4 cot θ csc θ ↔ which is the option A.
Question 2) Which polar equation is equivalent to the parametric equations below?
<span>
x=sin(theta)cos(theta)+cos(theta)
y=sin^2(theta)+sin(theta)</span>
Answer: option B) r = sinθ + 1
Explanation:
1) Polar coordinates ⇒ x = r cosθ, and y = r sinθ
2) replace x and y in the parametric equations:
a) r cosθ = sin(θ)cos(θ)+cos(θ)
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b) r sinθ =sin²(θ)+sin(θ)</span>
3) work both equations
a) r cosθ = sin(θ)cos(θ)+cos(θ) ⇒ r cosθ = cosθ [ sin θ + 1] ⇒ r = sinθ + 1
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b) r sinθ =sin²(θ)+sin(θ) ⇒ r sinθ = sinθ [sinθ + 1] ⇒ r = sinθ + 1
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</span><span>
</span>Therefore, the answer is r = sinθ + 1 which is the option B.
(x+12)(x-3) Try this answer hopefully it is correct.
Answer:
y=45
x=9
z= 9 root 2
Step-by-step explanation:
This is a right isosceles triangle, so
y=45
x=9
z= 9 root 2
9²+9²=z²
81+81=z²
162=z²
√162=z
z=9√2
Step-by-step explanation:
step 1. (13x + 8) + (2x + 7) = 180 (definition of same side interior angles)
step 2. 15x + 15 = 180 (like terms)
step 3. 15x = 165 (subtract 15 from each side)
step 4. x = 11. (divide both sides by 15)
F(x) = a(x-h)²<span> + k
</span><span><u>Given that the vertex is (-3 -6):</u>
</span>f(x) = a(x + 3)² -6
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<u>Given that it passes through (0.0), find a:</u>
a(0 + 3)</span>² - 6 = 0
<span>
9a - 6 = 0
9a = 6
a = 6/9 =2/3
<u>So the equation is :</u>
</span>f(x) = 2/3(x + 3)² -6
<span>
<u>Write the equation in standard form:</u>
</span>f(x) = 2/3(x² + 6x + 9) - 6
f(x) = 2/3x² + 4x + 6 - 6
f(x) = 2/3x² + 4x
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Answer: </span>f(x) = 2/3x² + 4x<span>
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