For each of the described curves, decide if the curve would be more easily given by a polar equation or a Cartesian equation. Then write an equation for the curve. (a) A line through the origin that makes an angle of π6 with the positive x-axis.
(b) A vertical line through the point (5,5).

(a) This expression would be more easily represented by a Cartesian equation, since distance with respect to origin varies with angle. (). The line is represented by .

(b) This expression would be more easily represented by a Cartesian equation, since distance with respect to origin varies with angle. (). The resulting equation is .

Step-by-step explanation:

(a) This expression would be more easily represented by a Cartesian equation, since distance with respect to origin varies with angle. () A straight line is represented by the following expression:

Where:

- Independent variable, dimensionless.

- Dependent variable, dimensionless.

- Slope, dimensionless.

- y-Intercept.

From Analytical Geometry we know that slope is equal to:

Where is the angle of the line with respect to the positive x-axis, measured in radians.

If we know that , then:

The y-intercept is found by replacing independent and dependent variables by known point:

Therefore, the line is represented by .

(b) This expression would be more easily represented by a Cartesian equation, since distance with respect to origin varies with angle. () A straight vertical line is represented by:

Since slope becomes indefined by applying the definition used in (a).

Since that line passes through (5, 5), the value of is 5. Then, the resulting equation is .