Prove that the polar equation of a planet's elliptical orbit is (1 – e²)a e cos 0' where e is the eccentricity and 2a is the length of the major axis. r = 1 Planet Sun: focus at pole 2a

Q: Which of the following Polar equation represents the following Curve -2 1. (a)r = 1- sine (b) r = 2+…

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Q: Find the equation of the tangent line to the circle r=4cos(t) at the point where t=PI/6. (a)Also…

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Q: 4. Convert the parametric equations into a single Cartesian equation and use this to sketch the…

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Q: 1. Let C be the polar curve given by r = 2+ sin 0, –T/4<0 <#/2. Sketch the curve. b. Indicate with…

A: Given : C: polar curve=r=2+sin θ,      -π4≤ θ≤π2 (a)   The  graph of the polar curve and  (b)…

Q: Use the identity cos 2θ = cos2 θ − sin2 θ to find a polar equation of the hyperbola x2 − y2 = 1.

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Q: Find a polar equation for the curve represented by the given Cartesian equation.

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Q: Find a polar equation in the form r= f(θ) of the hyperbola with eccentricity e= 4 such that one of…

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Q: 4. Given the polar equation r =1+sin 0. (a) Sketch the curve by first sketching it in rectangular…

A: Given: The polar equation is r=1+sinθ. To find: a) Sketch the curve by sketching the rectangular…

Q: Consider the curve C of equation (2x² – y + y³)² = a² + y? From the choices given, select the one…

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Q: Find the polar equation for a parabola with its focus at the origin and whose directrix is y = -a…

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Q: Find an approximate polar equation for the ellipticalorbit of the earth around the sun (at one…

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Q: 1. Let r(0) = 2sin(0) + cos(Ⓒ) be the polar definition of a planar curve. Find, in standard x-y…

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Q: The equation of the polar curve above is E. r = 3 + 2 sin 0 F. r = 3 - 2 sin 0 G. r = 3+ 2 cos 0 H.…

A: From the direction of the cardioid, extending horizontally, that you…

Q: Find the exact length of the polar curve r=e2(theta) , where 0 is less than or equal too theta is…

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Q: r= 1+ 2 sin e.. Roughly Sketch the polar curve

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Q: - The polar curve r= 5sin e has the eccentricity

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Q: The polar equation for the curve whose rectangular(cartesian)equation xy = 2 is: (a)rcosesine = 4…

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Q: 3) Sketch the polar curve r = 1 + 2 cos 0. In your graph, label the points corresponding to the…

A: See the attachment

Q: Find the length of the curve over the given interval. Polar Equation Interval r = 9 cos(8) 2

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Q: 1. Describe the graphs of the equation (a) r = cos 20 in cylindrical coordinates. (b) p= sec otan 0…

A: First change cylindrical and spherical coordinates system to Cartesian coordinates system

Q: Convert the polar equation 4r cos θ + r sin θ = 8 into a rectangular equation that expresses y in…

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Q: Find a polar equation for the curve represented by the given Cartesian equation. x² + y2 = 16

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Q: Find a polar equation for the curve represented by the given Cartesian equation. 4y? =

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Q: The curve is given in polar coordinates by r = e³0. Let A be the point on the curve corresponding to…

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Q: A curve with polar equation r = 8 sin 0 + 49 cos 0 represents a line. This line has a Cartesian…

A: The polar equation is, r=58sinθ+49cosθ   Rewrite the equation. 8rsinθ+49rcosθ=5   Substitute y for…

Q: 9. (a) Find the rectangular coordinates of the point(s) of the graph of the polar equation r = 4 sin…

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Q: Find the polar equation of a hyperbola with eccentricity 2, and directrix y = 5. Select the correct…

A: We have to find the polar equation of the hyperbola with eccentricity 2 and directrix y=5.

Q: Consider the polar equation of a conic. Determine the eccentricity. -5 r = 1 + cos(0) e =

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Q: Sketch the curve with the given polar equation by first sketching the graph of r as a function of θ…

A: We need to sketch the curve with the given polar equation by first sketching the graph of r as a…

Q: Q1) Find the polar equation of the circle whose center has polar coordinates (a, 0,) and radius a.…

A: Given a polar point (a, theta 0) and radius a.

Q: Find a polar equation of the form r^2=f(θ) for the curve represented by the Cartesian equation x^2 −…

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Q: Find a polar equation for the curve represented by the given Cartesian equation. xy=8

A: We havexy=8

Q: 9 Given the polar equation r = 3 + 4 cos(0) The eccentricity is e = The equation of the directrix…

A: Consider the given equation is, r=93+4cosθ Consider the general form is, r=ep1±ecosθ Then,…

Q: 4. Find a polar equation for the curve represented by the Cartesian equation y = 3x. T = r cos 0, y…

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Q: 1. Transform the polar equation 4 cos 0 cos² 0 + 1 into its Cartesian form and identify the curve.

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Q: A curve with polar equation r = 20 4 sin 0+ 17 cos 0 represents a line. This line has a Cartesian…

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Q: 10. Consider the curve with polar equation r = 6 sin 0 where 0< 0 <T. WVithout finding the Cartesian…

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Q: 36 r= e®,-π <θ <T

A: Consider a function f such that its derivative is continuous in the interval α≤θ≤β. Then the length…

Q: Consider the curve C of equation 4.xy x² + y? x² + y² From the choices given, select the one that…

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Q: b, The polar equation for the curve whose rectangular(cartesian)equation xy32 is: (a)rcosesine = 4 4…

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Q: Find the polar equation of the conic with focus at the origin and the given eccentricity and…

A: We know eccentricity is the Ratio of distance between variable point to directrix to distance…

Q: Sketch the graph of the polar equation.r = e2θ, θ ≥ 0 (logarithmic spiral)

A: Consider the given polar curve to be r=e2θ, θ≥0.

Q: (e) Graph the surface in R' whose spherical equation is p = cos 20.

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Q: Find the length of the curve r= /1+ cos 20, 0<0<n/2/6. The length of the polar curve is.

A: We know the arc length formula for polar coordinates is L=∫θ1θ2r2+drdθ2dθ. Here, r=1+cos(2θ),…

Q: The graph of the polar equation r(0) = 0 cos(0), o ses. with derivative = cos(e) – 0 sin(0), is…

A: “Since you have asked multiple questions, we will solve the first question for you. If you want any…

Q: Find the polar equation for the curve represented by the equation y = 1+ 3x. A 1 ,0) sin 0-3 cos 0 В…

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Q: Find the length of the spiraling polar curve r = 4e^6(theta) from 0 to 2pi

A: Length of polar curve

Q: (b) Find the graph of the parametric equations: I = sin 0, y = 3 – 2 cos 20

A: Plugging values of θ, to find the corresponding values of x and y: When…

Q: A curve with polar equation 32 3 sin 8 + 13 cos e represents a line. This line has a Cartesian…

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