Question

How do you identify the conic of ##r = 3/(1 - sinx)?

Answer 1

Parabola. See explanation.

Explanation:

Referred to the/a focus S as r =0 and the perpendicular from S to

the ( or corresponding ) directrix as the the initial line `theta = 0`,

the equation to a conic is

(`theta = pi/2` semi-chord length )/r

= 1 + (eccentricity of the conic ) `cos theta` or simply

`l/r = 1+ e cos theta`.

If the perpendicular is along `theta = alpha`, the equation

becomes

`l / r = 1+ e cos ( theta + alpha )`.

The conic is an ellipse, parabola or hyperbola according as

`e < = > 1`

Here,

`3/r = 1 - sin theta = 1 + (1 ) cos ( theta + pi/2 )`

`e = 1`. So, the conic is a parabola, with focus at S ( r = 0 ) and the

perpendicular from S to the directrix is along

`theta = alpha = pi/2`

The Cartesian equivalent is

`3 = r - r sin theta = sqrt( x^2 + y^2 ) - y`

The graph is immediate.
graph{3 - sqrt ( x^2 + y^2) + y = 0}

.

Answer 2

Parabola

Explanation:

Let’s see by first putting it in rectangular form:

`r(1-sinx)= 3`

`r-rsinx=3`

`r-y=3`

`r=y+3`

Square both sides:

`r^2= y^2+6y+9`

`x^2+y^2= y^2+6y+9`

`x^2= 6y+9`

`(x^2-9)/6= y`

`y= 1/6x^2-3/2`

Answer 3

Since `e=1` this is parabola , directrix is `3` unit below
focus (pole) and parallel to the polar axis.

Explanation:

The polar equation of conic is `r= (ed)/(1-esin (x)` when

directrix is below the pole.

`r= 3 /(1-sin x) :. e=1 ,d=3` If `e = 1`, then the conic is

a parabola. If `e < 1`, then the conic is an ellipse.

If #e > 1=, then the conic is a hyperbola.

Since `e=1` this is parabola , directrix is `3` unit below

focus (pole) and parallel to the polar axis. [Ans]