Question

How do you approach and solve this problem?

A curve called a witch of Maria Agnesi consists of all possible positions of the point P in the figure. Show that parametric equations for this curve can be written as
`x=2acottheta`
`y=2asin^2theta`
Sketch the curve.

Answer 1

Please see below.

Explanation:

As point `A` moves anticlockwise starting from point `O`, we observe that angle `theta` changes from `0` to `2pi`. Further, coordinates of `P(x,y)` depend on one - `a` the radius of the circle, which is a constant and two - `theta`, which we have seen as varying from `0` to `2pi`. Hence, we ought to work out for `x` and `y`, using these.

For this purpose and to explain it better, I intend to draw a perpendicular from `PN` to `x`-axis. and also name the diameter as `OL` and perpendicular `AM` on `x`-axis#. Also observe that

`m/_OLA=m/_LCO=theta`

Now `x=ON=NCxxcottheta=2acottheta`

and `y=PN=AM=OAsintheta`

but in `DeltaOLA`, `(OA)/(OL)=sin/_OLA=sintheta`

Therefore `y=OAsin^2theta=2asin^2theta`

Hence, parametric form of all possible positions of the point `P` is

`(2acottheta,2asin^2theta)`

We can also find locus in rectangular coordinates too by eliminating `theta` from these equations.

Observe that `x^2=4a^2cot^2theta`

= `4a^2(csc^2theta-1)`

= `4a^2(1/sin^2theta-1)`

i.e. `x^2=4a^2((2a)/y-1)`

and if `a=5` this becomes `x^2=1000/y-100` and graph (includes the circle too) appears as

graph{(1000/y-100-x^2)(x^2+y^2-10y)=0 [-22.5, 22.5, -7.71, 14.79]}