Problem source

The curve $C_1$ has parametric equations $x=t^2$, $y= t^3$, where 
$-\infty < t < \infty\,$. 
Let $O$ denote the point $(0,0)$.
The points $P$ and $Q$ on $C_1$  are such that $\angle POQ$ is
a right angle. Show that the tangents to $C_1$  at $P$ and $Q$ 
intersect on the curve $C_2$  with equation $4y^2=3x-1$.

Determine whether  $C_1$ and $C_2$ meet, and sketch the two
curves on the same axes.

Solution source

$\angle POQ = 90^\circ$ means that if $P(p^2,p^3)$ and $Q(q^2,q^3)$ are our points then $OP^2+OQ^2 = PQ^2$, so

\begin{align*}
&& p^4+p^6+q^4+q^6 &= (p^2-q^2)^2+(p^3-q^3)^2 \\
&&&= p^4+q^4-2p^2q^2+p^6+q^6-2p^3q^3 \\
\Rightarrow && 0 &= 2p^2q^2(1+pq) \\
\Rightarrow && pq &= -1 \\
\\
&& \frac{\d y}{ \d x} &= \frac{\frac{\d y }{\d t}}{\frac{\d x}{\d t}} \\
&&&= \frac{3t^2}{2t} = \tfrac32t \\
\Rightarrow && \frac{y-p^3}{x-p^2} &= \tfrac32p \\
\Rightarrow && 2(y-p^3) &=3p(x-p^2) \\
&&  2(y-q^3) &=3q(x-q^2) \\
\Rightarrow && 2(q^3-p^3) &= (3p-3q)x+3(q^3-p^3) \\
&& p^3-q^3 &= 3(p-q)x \\
\Rightarrow && x &= \tfrac13(p^2+q^2+pq) \\
&& 2y &= 3p(\tfrac13(p^2+q^2+pq)-p^2)+2p^3 \\
&&&= p(p^2+q^2+pq)-p^3 \\
&&&= pq^2+p^2q \\
&&&= -p-q \\
&&y&= -\frac{p+q}{2} \\
\\
&& 4y^2 &= p^2+q^2 \\
&& 3x-1 &= p^2+q^2 \\
\end{align*}

To check if they meet, try $4t^6=3t^2 - 1$.

Consider $y = 4x^3-3x+1$ $y(0) = 1$ and $y' = 12x^2-3 = 3(4x^2-1)$ which has roots at $\pm \tfrac12$, therefore we need to test $y(\tfrac12) = \tfrac12-\tfrac32 + 1 = 0$, so there is a one intersection at $x = \tfrac1{2}, y = \tfrac1{2\sqrt{2}}$


\begin{center}
    \begin{tikzpicture}
    % Slightly expanded bounding limits for breathing room
    \def\xl{-.3}; \def\xu{2.3};
    \def\yl{-1.6}; \def\yu{1.6};
    
    % Calculate scaling factors to make the plot square
    \pgfmathsetmacro{\xrange}{\xu-\xl}
    \pgfmathsetmacro{\yrange}{\yu-\yl}
    \pgfmathsetmacro{\xscale}{10/\xrange}
    \pgfmathsetmacro{\yscale}{10/\yrange}
    
    % Define the reusable styles to keep code clean
    \tikzset{
        x=\xscale cm, y=\yscale cm,
        axis/.style={thick, draw=black!80, -{Stealth[scale=1.2]}},
        grid/.style={thin, dashed, gray!30},
        curveA/.style={very thick, color=cyan!70!black, smooth},
        curveB/.style={very thick, color=orange!90!black, smooth},
        dot/.style={circle, fill=black, inner sep=1.2pt},
        labelbox/.style={fill=white, inner sep=2pt, rounded corners=2pt} % Protects text from lines
    }
    
    % Draw background grid
    \draw[grid] (\xl,\yl) grid[step=0.5] (\xu,\yu);
    
    % Set up axes
    \draw[axis] (\xl,0) -- (\xu,0) node[right, black] {$x$};
    \draw[axis] (0,\yl) -- (0,\yu) node[above, black] {$y$};
    
    % Define the bounding region with clip
    \begin{scope}
        \clip (\xl,\yl) rectangle (\xu,\yu);
        
        \draw[curveA, domain=-3:3, samples=150] 
            plot ({\x*\x},{\x*\x*\x});
            
        \draw[curveB, domain=-3:3, samples=150] 
            plot ({(4*\x*\x+1)/3},{\x});
    \end{scope}
    
    % Annotate Function Names
    \node[curveA, labelbox] at (1.2, 1.25) {$y^2 = x^3$};
    \node[curveB, labelbox] at (1.85, -1.1) {$x = \frac{4y^2+1}{3}$};
    
    % Define specific coordinates mathematically to avoid repeating math expressions
    \def\px{0.5}
    \def\py{0.35355} % 1/(2*sqrt(2))
    
    % Top intersection
    \node[dot] at (\px, \py) {};
    \node[labelbox, anchor=south east] at (\px - 0.03, \py + 0.05) {$(\frac{1}{2}, \frac{1}{2\sqrt{2}})$};
    
    % Bottom intersection
    \node[dot] at (\px, -\py) {};
    \node[labelbox, anchor=north east] at (\px - 0.03, -\py - 0.05) {$(\frac{1}{2}, -\frac{1}{2\sqrt{2}})$};
    
    \end{tikzpicture}
\end{center}