Question

# 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

Conic sections
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. $$e = 2,\ x = 4$$

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. $$r=\ \frac{2(4)}{1\ +\ 2\ \cos\ \theta}$$
$$r=\ \frac{8}{1\ +\ 2\ \cos\ \theta}$$