<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] (iii) 
You have an error here. 
That the normal to H at the point $T$ meets $H$ again at $P(cp , \frac{c}{p})$ means that the normal to $H$ at the point $T$ is the line $TP$.
The gradient of the line $TP$ is given by
$$\frac{c/p-c/t}{cp-ct}=\frac{-(p-t)}{(p-t)pt}=-\frac{1}{pt}$$
Since this is equal to $t^2$, we have
$$-\frac{1}{pt}=t^2\quad\Rightarrow\quad pt^3=-1$$
(iv) 
Here, you have to consider the coordinates of $M$, not the equation of the line $TP$.
Let $M(X,Y)$.
Then, using $p=-1/t^3$,
$$X=\frac{cp+ct}{2}=\frac{-c/t^3+ct}{2}=\frac{c(t^4-1)}{2t^3}$$
$$Y=\frac{c/p+c/t}{2}=\frac{-ct^3+c/t}{2}=\frac{c(1-t^4)}{2t}$$
Then, 
$$\begin{align}&4X^3Y^3+c^2(X^2-Y^2)^2\\&=4\left(\frac{c(t^4-1)}{2t^3}\right)^3\left(\frac{c(1-t^4)}{2t}\right)^3+c^2\left(\left(\frac{c(t^4-1)}{2t^3}\right)^2-\left(\frac{c(1-t^4)}{2t}\right)^2\right)^2\\&=\frac{-c^6(t^4-1)^6}{16t^{12}}+\frac{c^2\cdot c^4(1-t^4)^6}{16t^{12}}\\&=0\end{align}$$
</s>