<s>[INST] Conic Sections: Hyperbola (Finding the Locus) This is a multipart question so bear with me until I get to the part where I am stuck on.


$H$: $xy=c^2$ is a hyperbola
(i) Show that $H$ can be represented by the parametric equations $x=ct$ , $y= \frac{c}{t}$.


If we take $y= \frac{c}{t}$ and rearrange it to $t= \frac{c}{y}$ and subbing this into $x=ct$
$$ x = c(\frac{c}{y}) $$
$$ \therefore xy = c^2 $$


(ii) Find the gradient of the normal to $H$ at the point $T$ with the coordinates $(ct, \frac{c}{t})$


As $xy = c^2$
$$ \Leftrightarrow y = c^2 x^{-1} $$
$$ \Leftrightarrow y' = -\frac{c^2}{x^2} $$
$$ \Leftrightarrow y'_{x=ct} = \frac{-1}{t^2} $$
Hence gradient of the normal is $m = t^2$


The normal to H at the point $T$ meets $H$ again at $P(cp , \frac{c}{p})$
(iiI) By finding the gradient of the line TP and comparing it with the gradient from (ii), find a relationship between $t$ and $p$ 


So repeating steps of (ii) again 
$xy = c^2$
$$ \Leftrightarrow y = c^2 x^{-1} $$
$$ \Leftrightarrow y' = -\frac{c^2}{x^2} $$
$$ \Leftrightarrow y'_{x=cp} = \frac{-1}{p^2} $$
Hence gradient of the normal is $$m = p^2$
Comparing gradients 
$$ p^2 = t^2 $$
$$ \Leftrightarrow |p| = |t| $$ Now do we assume $t>0$ and $p>0$? which gives us
$$ \Leftrightarrow p = t $$ as our relationship.
Now this is where I am mostly stuck on


(iv) Hence or otherwise by finding the parametric equations of $M$, the midpoint of $TP$ in terms of $t$, show that the coordinates of $M$ lie on the curve $4x^3y^3 + c^2(x^2-y^2)^2 = 0$ 
Note: This can also be stated as "Find the Locus of M"


So basically what I have done is to find the normal line of TP
Which is given as 
$$ (y-y_1)=m(x-x_1) $$
Because they want everything in terms of $t$ , let $p=t$
$$ (y-\frac{c}{t}) = t^2(x-ct) $$
$$ \Leftrightarrow (y-\frac{c}{t}) = t^2x - ct^3 $$ 
$$ \Leftrightarrow y = t^2x - ct^3 + \frac{c}{t} $$
Now this is stuck how should I continue?
 [/INST] Part (iii):
As pointed out by @mathlove, $$p=-1/t^3$$.
Part (iv): Coordinates of $M$ are given by 
$$x=\frac 12 \left(ct-\frac c{t^3}\right)\qquad\Rightarrow \frac {2x}c=t-\frac 1{t^3}=X\\
y=\frac 12 \left(\frac ct-ct^3\right)\qquad\Rightarrow \frac {2y}c=\frac 1t-t^3=Y\\\\
\frac XY=-\frac 1{t^2}\\
XY=2-t^4-\frac 1{t^4}=-\left(t^2-\frac 1{t^2}\right)^2=-\left(-\frac YX+\frac XY\right)^2\\
(XY)^3=-(X^2-Y^2)^2\\
\left(\frac {2x}c\cdot \frac {2y}c\right)^3=-\left(\left(\frac{2x}c\right)^2-\left(\frac{2y}c\right)^2\right)^2\\
4x^3y^3=-c^2(x^2-y^2)^2\qquad\blacksquare$$
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