Write the equation 5x – 4y = –7 in polar form
To write the equation 5x - 4y = -7 in polar form, we need to express x and y in terms of polar coordinates.
In Cartesian coordinates, x and y represent the coordinates of a point on a plane. In polar coordinates, a point is represented by its distance from the origin (r) and its angle (θ) with respect to the positive x-axis.
We can use the following relationships to convert between Cartesian and polar coordinates:
x = r * cos(θ)
y = r * sin(θ)
To write the equation in polar form, we'll substitute these expressions into the equation 5x - 4y = -7:
5(r * cos(θ)) - 4(r * sin(θ)) = -7
Now, let's simplify the equation:
5r * cos(θ) - 4r * sin(θ) = -7
Dividing both sides of the equation by r:
5 * cos(θ) - 4 * sin(θ) = -7/r
The expression -7/r can be simplified, as r ≠ 0 in polar coordinates. Let's say -7/r = k:
5 * cos(θ) - 4 * sin(θ) = k
This equation is the polar form of the given equation 5x - 4y = -7. The parameter k represents the scaled distance from the origin.