3. Let $O , P , P _ { 1 } , P _ { 2 }$ be the points in the $( x , y )$-plane with coordinates $( 0,0 ) , ( s , 1 / s )$, $\left( s _ { 1 } , 1 / s _ { 1 } \right) , \left( s _ { 2 } , 1 / s _ { 2 } \right)$ respectively. (i) Using the axes below, sketch the curve traced out by $P$ as $s$ varies over non-zero real values, and find an equation for the curve in the form $y = f ( x )$. (ii) Write down the equation of the straight line $P P _ { 1 }$ joining $P$ to $P _ { 1 }$, giving your answer in the form $y = m _ { 1 } x + c _ { 1 }$. (iii) Show that the line $P P _ { 1 }$ is perpendicular to $P P _ { 2 }$ if, and only if, $s _ { 1 } s _ { 2 } = - 1 / s ^ { 2 }$. (iv) Let $m _ { 1 } , m _ { 2 } , n _ { 1 } , n _ { 2 }$ be the gradients of the lines $P P _ { 1 } , P P _ { 2 } , O P _ { 1 } , O P _ { 2 }$ respectively. Show that $$\left( \frac { m _ { 1 } } { m _ { 2 } } \right) ^ { 2 } = \frac { n _ { 1 } } { n _ { 2 } }$$ [Figure]
3. Let $O , P , P _ { 1 } , P _ { 2 }$ be the points in the $( x , y )$-plane with coordinates $( 0,0 ) , ( s , 1 / s )$, $\left( s _ { 1 } , 1 / s _ { 1 } \right) , \left( s _ { 2 } , 1 / s _ { 2 } \right)$ respectively.\\
(i) Using the axes below, sketch the curve traced out by $P$ as $s$ varies over non-zero real values, and find an equation for the curve in the form $y = f ( x )$.\\
(ii) Write down the equation of the straight line $P P _ { 1 }$ joining $P$ to $P _ { 1 }$, giving your answer in the form $y = m _ { 1 } x + c _ { 1 }$.\\
(iii) Show that the line $P P _ { 1 }$ is perpendicular to $P P _ { 2 }$ if, and only if, $s _ { 1 } s _ { 2 } = - 1 / s ^ { 2 }$.\\
(iv) Let $m _ { 1 } , m _ { 2 } , n _ { 1 } , n _ { 2 }$ be the gradients of the lines $P P _ { 1 } , P P _ { 2 } , O P _ { 1 } , O P _ { 2 }$ respectively. Show that
$$\left( \frac { m _ { 1 } } { m _ { 2 } } \right) ^ { 2 } = \frac { n _ { 1 } } { n _ { 2 } }$$
\includegraphics[max width=\textwidth, alt={}, center]{2943001e-0059-4191-9307-3bd89df4b942-10_928_939_1062_571}\\