Convert the
given polar equation to a Cartesian equation.
r^2=4sec(theta)csc(theta)
in your repertoire of relations in this topic you should have:
x = rcosØ and y = rsinØ, and r^2 = x^2 + y^2
then r/x = sec‚ and r/y = cscØ
so let's just substitute
r^2 = (4r/x)(r/y)
r^2 = 4r^2 /(xy)
1 = 4/(xy)
xy = 4
To convert a polar equation to a Cartesian equation, we need to use the relationships between polar and Cartesian coordinates. Here's how we can do it step by step:
Step 1: Start with the given polar equation.
r^2 = 4sec(theta) * csc(theta)
Step 2: Recall that the relationships between polar and Cartesian coordinates are:
x = r * cos(theta)
y = r * sin(theta)
Step 3: Square both sides of the equation to eliminate the square on r:
r^2 = 4sec(theta) * csc(theta)
(r * r) = 4sec(theta) * csc(theta)
Step 4: Replace r^2 with (x^2 + y^2) and rewrite sec(theta) and csc(theta) in terms of cos(theta) and sin(theta):
(x^2 + y^2) = 4(1/cos(theta))(1/sin(theta))
Step 5: Simplify by multiplying 4 with 1/cos(theta) and 1/sin(theta):
(x^2 + y^2) = 4 * (1/(cos(theta) * sin(theta)))
Step 6: Combine the fractions:
(x^2 + y^2) = 4/(cos(theta) * sin(theta))
Step 7: To further simplify the equation, recall the trigonometric identity:
sin(theta) * cos(theta) = 1/2 * sin(2theta)
Using the identity, we can rewrite the equation as:
(x^2 + y^2) = 8/sin(2theta)
Step 8: Finally, divide both sides of the equation by sin(2theta) to isolate x^2 + y^2:
(x^2 + y^2) / sin(2theta) = 8
And that's the Cartesian equation for the given polar equation r^2 = 4sec(theta) * csc(theta).