Convert 2xy = 1 to polar form.
To convert the equation 2xy = 1 to polar form, we need to express the variables x and y in terms of r and theta. In polar form, a point (x, y) is represented as (r, theta), where r is the distance from the origin to the point, and theta is the angle between the positive x-axis and the line segment connecting the origin to the point.
Step 1: Express x in terms of r and theta
Using the relationship between polar and rectangular coordinates, we have:
x = r * cos(theta)
Step 2: Express y in terms of r and theta
Similarly,
y = r * sin(theta)
Now, substitute these expressions for x and y back into the original equation:
2xy = 1
2(r * cos(theta))(r * sin(theta)) = 1
Simplify the equation:
2r^2 * cos(theta) * sin(theta) = 1
This is the equation in polar form.
To convert the equation 2xy = 1 to polar form, we need to express x and y in terms of polar coordinates.
In polar form, x = r * cos(theta) and y = r * sin(theta), where r is the distance from the origin to the point and theta is the angle that the line connecting the point to the origin makes with the positive x-axis.
Substituting these values into the given equation, we get:
2 * (r * cos(theta)) * (r * sin(theta)) = 1
Simplifying this equation, we have:
2r^2 * cos(theta) * sin(theta) = 1
Since cos(theta) * sin(theta) = 1/2 * sin(2theta), we can rewrite the equation as:
r^2 * sin(2theta) = 1/2
Finally, dividing both sides of the equation by sin(2theta), we get the polar form:
r^2 = 1 / (2 * sin(2theta))
2 (r cos T)(r sin T) = 1
r^2 cos T sin T = (1/2)