Transform the polar equation to an equation in rectangular coordinates. Then identify and graph the equation. r cos \theta = 4 What is the standard form of the equation in rectangular form? What is the graph of this equation? A. vertical line C. circle with center at (0, 4) Select the graph of r cos \theta = 4. B. horizontal line D. circle with center at (4, 0)
The Correct Answer and Explanation is:
To solve the equation rcosθ=4r \cos \theta = 4rcosθ=4 and transform it into rectangular coordinates, we need to use the relationship between polar and rectangular coordinates. The polar coordinates (r,θ)(r, \theta)(r,θ) are related to rectangular coordinates (x,y)(x, y)(x,y) by the following formulas: x=rcosθandy=rsinθx = r \cos \theta \quad \text{and} \quad y = r \sin \thetax=rcosθandy=rsinθ
Given the equation rcosθ=4r \cos \theta = 4rcosθ=4, we can replace rcosθr \cos \thetarcosθ with xxx (since x=rcosθx = r \cos \thetax=rcosθ), so the equation becomes: x=4x = 4x=4
This is the equation of a vertical line in rectangular coordinates, where the x-coordinate is always 4, and the y-coordinate can be any value.
Standard Form:
The standard form of this equation is simply: x=4x = 4x=4
Graph:
The graph of the equation x=4x = 4x=4 is a vertical line that passes through the point (4,0)(4, 0)(4,0) on the x-axis, extending infinitely upwards and downwards. This corresponds to the vertical line option.
Answer to Graph Selection:
- The correct graph would be A (a vertical line).
Thus, the correct standard form of the equation in rectangular coordinates is x=4x = 4x=4, and the graph corresponds to a vertical line, which is represented by option A.
