(a) Given the conic section displaystyle{r}=frac{5}{{{7}+{3} cos{{left(thetaright)}}}}, find the x and y intercept(s) and the focus(foci). (b) Given the conic section displaystyle{r}=frac{5}{{{2}+{5} sin{{left(thetaright)}}}}, find the x and y intercept(s) and the focus(foci).

Carol Gates 2020-12-16 Answered
(a) Given the conic section r=57+3cos(θ), find the x and y intercept(s) and the focus(foci).
(b) Given the conic section r=52+5sin(θ), find the x and y intercept(s) and the focus(foci).
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Expert Answer

Nola Robson
Answered 2020-12-17 Author has 94 answers
(a)
Given r=57+3cosθ
It can be written as r=5/71+3/7cosθ
Comparing it with general conic section r=ed1+ecosθ, we get
e=37, since e i.e. eccentricity lies between 0 & 1. :. given conic section is ellipse.
for x-intercept θ=0,
π when θ=0,r=510=12, so x-intercept in this case is (1/2, 0)
when θ=π.r=54, so x-intercept in this case is (54,0)
for y-intercept θ=π2,3π2
when θ=π2,r=57, so y-intercept in this case is (0,57)
when θ=3π2,r=57, so y-intercept in this case is (0,57)
One foci is (0, 0) of this form. Now, to find other foci we will find out center of the ellipse & it will be the mid point of both the x-intercepts
i.e. center =(12542,0)
=(38,0)
Now we will distance of center (38,0) and 1st foci (0,0),c=3/8
since the distance of both the foci from the center is same, so coordinates of 2nd foci is (3838,0)
i.e. (68,0)
so, x intercepts are (12,0),(54,0)
y intercepts are (0,57),(0,57)
foci are (0,0),(68,0)
(b)
Given r=52+5sinθ
It can be written as r=5/21+5/2sinθ
Comparing it with general conic section r=ed1+esinθ, we get
e=52, since e i.e. eccentricity is greater than 1. given conic section is hyperbola.
for x-intercept θ=0,π
when θ=0,r=52, so x -intercept in this case is (52,0)
when θ=π.r=52,so x -intercept in this case is (52,0)
for y-intercept θ=π2,3π2
when θ=π2,r=57, so y-intercept in this case is (0,57)
when

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