Final answer:
The curve given by the polar equation r = 7cos(θ) can be converted to a Cartesian equation representing a circle with a radius of 7/2 and centered at (7/2, 0) in the Cartesian plane.
Step-by-step explanation:
To find a Cartesian equation for the curve given by the polar equation r = 7cos(θ), we'll convert the polar coordinates to Cartesian coordinates using the conversion formulas:
- x = r × cos(θ)
- y = r × sin(θ)
Substitute r = 7cos(θ) into the first formula to get:
x = 7cos(θ) × cos(θ) = 7cos^2(θ)
Now, use the trigonometric identity cos^2(θ) = (1 + cos(2θ))/2:
x = 7 × (1 + cos(2θ))/2
This simplifies to:
x = (7/2) + (7/2)cos(2θ)
Using the second formula, we can then express cos(2θ) in terms of x and y, by noting that cos(2θ) = 2cos^2(θ) - 1 and sin^2(θ) = 1 - cos^2(θ), giving us y^2 = r^2sin^2(θ) = 49(1 - cos^2(θ)) = 49 - x^2.
Finally, we rearrange terms to find the Cartesian equation:
x = (7/2) + (7/4)(1 - y^2/49)
By identifying the form, we can see that it's an equation of a circle with radius 7/2 and centered at (7/2, 0) on the Cartesian plane.