The conic section represented by the polar equation r = 6 / (3 - 4 cos(theta)) is an ellipse.
To find the eccentricity, orientation, and directrix of the ellipse, we first need to rewrite the equation in the standard form of an ellipse equation in polar coordinates, which is given by:
r = (a * (1 - e^2)) / (1 + e * cos(theta))
Where:
a = semi-major axis
e = eccentricity
Comparing the given equation r = 6 / (3 - 4 cos(theta)) to the standard form, we can see that a = 6, e = 4/3. Therefore, the eccentricity of the ellipse is given by:
e = sqrt(1 - (b^2/a^2))
Therefore, the eccentricity is:
e = sqrt(1 - (4^2 / 6^2)) = sqrt(1 - 16/36) = sqrt(20/36) = 2/3
The orientation of the ellipse can be determined by the angle of the major axis with respect to the positive x-axis:
tan(theta) = (b / a) * sqrt(1 - e^2)
tan(theta) = (3 / 6) * sqrt(1 - (4/3)^2) = 0.5 * sqrt(1 - 16/9) = 0.5 * sqrt(9/9 - 16/9) = 0.5 * sqrt(-7/9)
Therefore, the orientation of the ellipse is given by:
theta = arctan(0.5 * sqrt(-7/9))
Finally, the directrix of the ellipse can be found by the equation:
r = (a * (1 - e^2)) / (1 + e * cos(theta))
Therefore, the equation of the directrix is given by:
r = (6 * (1 - (4/3)^2)) / (1 + (4/3) * cos(theta)) = (6 * (1 - 16/9)) / (1 + (4/3) * cos(theta)) = (6 * 1/9) / (1 + (4/3) * cos(theta)) = 2/3 / (1 + (4/3) * cos(theta))
Hence, the equation of the directrix is r = 2 / (3 + 4cos(theta))
Which conic section is represented by r=6/ 3 - 4 cos theta.
What is it's eccentricity, orientation, and directrex
1 answer
1 answer