Question

Transform each polar equation to an equation in rectangular coordinates and identify its shape.

Answer 1

Answer:

(a) $x ^ 2 + y ^ 2 = 6 ^ 2$

(b) $(x-1) ^ 2 + y ^ 2 = 1$

Step-by-step explanation:

Remember that to convert from polar to rectangular coordinates you must use the relationship:

$x = rcos(\theta)$

$y = rsin(\theta)$

$x ^ 2 + y ^ 2 = r ^ 2$

In this case we have the following equations in polar coordinates.

(a) $r = 6$.

Note that in this equation the radius is constant, it does not  depend on $\theta$.

As $r ^ 2 = x ^ 2 + y ^ 2$

Then we replace the value of the radius in the equation and we have to::

$x ^ 2 + y ^ 2 = 6 ^ 2$

Then $r = 6$ in rectangular coordinates is a circle centered on the point (0,0) and with a constant radius $r = 6$.

(b) $r = 2cos(\theta)$

The radius is not constant, the radius depends on $\theta$.

To convert this equation to rectangular coordinates we write

$r = 2cos(\theta)$     Multiply both sides of the equality by r.

$r ^ 2 = 2 *rcos(\theta)$    remember that $x = rcos(\theta)$, then:

$r ^ 2 = 2x$        remember that $x ^ 2 + y ^ 2 = r ^ 2$, then:

$x ^ 2 + y ^ 2 = 2x$         Simplify the expression.

$x ^ 2 -2x + y ^ 2 = 0$     Complete the square.

$x ^ 2 -2x + 1 + y ^ 2 = 1$

$(x-1) ^ 2 + y ^ 2 = 1$  It is a circle centered on the point (1, 0) and with radio $r=1$