Question

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 π/3 with the positive x-axis.
(b) A vertical line through the point (4, 4).

Answer 1

Answer:

a) $y=\sqrt{3}\ (x)$ is the equation of the curve that makes an angle π/3.

b) $x=3$ is the equation of line through the point (4,4).

Step-by-step explanation:

Given:

A line from origin which makes an angle of $\frac{\pi }{3}$ with x-axis.

A vertical line from $(4,4)$ .

We have to write the equation of the curves in Polar or Cartesian format.

Step wise:

a) A line from origin which makes an angle of $\frac{\pi }{3}$ with x-axis.

To write the equation of the above line in Polar coordinates is more desirable as the angles could be defined well in polar form.

So,

⇒ $y=mx$ ...equation (i)

⇒ $m=\frac{y}{x}$...here $m$ is the slope

The slope in terms of $\theta$ (angle) can be written as,

⇒ $tan(\theta)=\frac{y}{x}$

Plugging the values of the angle,$\theta =\frac{\pi }{3}$ .

⇒ $tan(\theta) =\frac{\pi}{3} = \sqrt{3}$ ...equation (ii)

Now re-arranging the equation (i) we can write it as,

⇒  $y=\sqrt{3}\ (x)$

b) A vertical line from $(4,4)$ .

Note:

The equation of a vertical line always takes the form x = k, where k is any number and k is also the x-intercept .

To write the above point in Cartesian coordinate is more acceptable and easy for us.

⇒ $x=4$

Then,

y = sq-rt(3) x is the equation of the curve that makes an angle π/3.

and x = 3 is the equation of line through the point (4,4).