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

### Relevant Questions

The eccentricities of conic sections with one focus at the origin and the directrix corresponding and sketch a graph:
$$\displaystyle{e}={2},$$
$$directrix \displaystyle{r}=-{2} \sec{\theta}.$$
Write a conic section with polar equation the focus at the origin and the given data hyperbola, eccentricity 2.5, directrix y = 2
Solve, a. If necessary, write the equation in one of the standard forms for a conic in polar coordinates $$r = \frac{6}{2 + sin \theta}$$ b. Determine values for e and p. Use the value of e to identify the conic section. c. Graph the given polar equation.
The polar equation of the conic with the given eccentricity and directrix and focus at origin: $$\displaystyle{r}={41}\ +\ {\cos{\theta}}$$
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Write the equation of each conic section, given the following characteristics:
a) Write the equation of an ellipse with center at (3, 2) and horizontal major axis with length 8. The minor axis is 6 units long.
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