Let the polar coordinates of the point (x, y) be (r, θ). Determine the polar coordinates for the point (2x, y)
For (x,y), we have
r^2 = x^2 + y^2
tanθ = y/x
Let the new point have polar coordinates (p,φ). Then
p^2 = (2x)^2 + y^2 = 4x^2 + y^2 = r^2 + 3x^2
tanφ = y/(2x) = 1/2 (y/x) = 1/2 tanθ
You can massage those into other forms, of course.
To determine the polar coordinates for the point (2x, y), we need to find the polar radius (r) and the polar angle (θ).
The polar radius (r) can be calculated using the distance formula:
r = √(x^2 + y^2)
Since we have the point (2x, y), we can substitute the values into the formula:
r = √((2x)^2 + y^2)
= √(4x^2 + y^2)
Next, we need to find the polar angle (θ). The polar angle represents the counterclockwise angle from the positive x-axis to the line segment connecting the origin to the point (2x, y).
The polar angle (θ) can be calculated using the inverse tangent (arctan) function:
θ = arctan(y / (2x))
Now we have the polar radius (r) and the polar angle (θ), which gives us the polar coordinates for the point (2x, y).
To find the polar coordinates for the point (2x, y), we need to convert the Cartesian coordinates (2x, y) to polar coordinates (r, θ).
Step 1: Calculate the radius, r:
The radius, r, can be found using the equation r = sqrt(x^2 + y^2). In this case, x = 2x and y = y, so the equation becomes:
r = sqrt((2x)^2 + y^2)
= sqrt(4x^2 + y^2)
Step 2: Calculate the angle, θ:
The angle, θ, can be found using the equation tan(θ) = y/x. In this case, x = 2x and y = y, so the equation becomes:
tan(θ) = y/(2x)
Step 3: Solve for θ:
To solve for θ, we can take the inverse tangent (tan^(-1)) of both sides:
θ = tan^(-1)(y/(2x))
Therefore, the polar coordinates for the point (2x, y) are (r, θ) = (sqrt(4x^2 + y^2), tan^(-1)(y/(2x))).