(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).

Question

(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).

Nola Robson · Accepted Answer

(a)  Given r=57+3cos⁡θ  It can be written as r=5/71+3/7cos⁡θ  Comparing it with general conic section r=ed1+ecos⁡θ, we get  e=37, since e i.e. eccentricity lies between 0 &amp;amp; 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 &amp;amp; it will be the mid point of  both the  x-intercepts  i.e. center =(12−542,0)  =(−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 (−38−38,0)  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+5sin⁡θ  It can be written as r=5/21+5/2sin⁡θ  Comparing it with general conic section r=ed1+esin⁡θ, 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

(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).

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