Find the polar equation of the conic with focus at the pole, having eccentricity 5 and
directrix y = -6.
Thank you so Much!
recall that r = ed/(1-e cosθ) has
eccentricity e and directrix x = d
so, r = 5*6/(1-5cosθ)
has directrix x = 6
But your directrix is that line rotated 3π/2, so
r = 30/(1-5cos(θ-3π/2)) = 30/(1+5sinθ)
To find the polar equation of the conic with focus at the pole and a given eccentricity and directrix, we can follow these steps:
Step 1: Identify the type of conic. Given that the focus is at the pole, we know that the conic is either a circle or an ellipse.
Step 2: Find the equation of the directrix in polar form. The directrix is given as y = -6, which can be converted to polar form using the equation x = r * cos(theta) and y = r * sin(theta). In this case, the equation becomes r * sin(theta) = -6.
Step 3: Calculate the distance from the focus to the directrix. The distance between the focus and the directrix is equal to the eccentricity (e) times the perpendicular distance between the origin and the directrix. In this case, the eccentricity is given as 5, so the distance is 5 * |-6| = 30.
Step 4: Determine the equation of the conic.
- If the distance from the focus to the directrix (30) is greater than the distance from the origin to the focus (r), then the conic is an ellipse.
- If the distances are equal, the conic is a parabola.
- If the distance from the focus to the directrix is less than the distance from the origin to the focus, the conic is a hyperbola.
Since the distance from the origin to the directrix is 6 and the distance from the origin to the focus is 30, we can see that 30 > 6. Therefore, the conic is an ellipse.
Step 5: Obtain the polar equation of an ellipse with a focus at the origin. For an ellipse with a focus at the origin, the polar equation is given by:
r = (d / (1 ± e * cos(theta)))
Where r is the distance from the origin to a point on the ellipse, d is the perpendicular distance from the origin to either the directrix or the major axis, e is the eccentricity, and theta is the angle between the polar axis (positive x-axis) and the line connecting the origin and the point on the ellipse.
In our case, the equation becomes:
r = (30 / (1 ± 5 * cos(theta)))
Note that the positive and negative signs in the equation correspond to two separate ellipses. The positive sign corresponds to one ellipse, and the negative sign corresponds to the other.
Therefore, the polar equation of the conic with a focus at the pole, eccentricity 5, and directrix y = -6 is:
r = (30 / (1 ± 5 * cos(theta)))
Hope this clarifies the solution for you!