Give the eccentricities of conic sections with one focus at the origin along with the directrix corresponding to that focus. Find a polar equation for each conic section. e = 2, x = 4

Question
Conic sections
asked 2021-02-25
Give the eccentricities of conic sections with one focus at the origin along with the directrix corresponding to that focus. Find a polar equation for each conic section. \(e = 2,\ x = 4\)

Answers (1)

2021-02-26
Step 1 Consider the given: eccentricity, \(e = 2\) directrix, \(x = 4\) If the eccentricity is greater than 1 then we get a hyperbola. And if the directrix is the line \(x = d,\ \text{then we have,}\ r=\ \frac{ed}{1\ +\ e\ \cos\ \theta}\) Step 2 According to question: Substitute the value of eccentricity and directrix in polar equation of hyperbola. \(r=\ \frac{2(4)}{1\ +\ 2\ \cos\ \theta}\)
\(r=\ \frac{8}{1\ +\ 2\ \cos\ \theta}\)
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