Challenging +1.2 This is a standard FP1 rectangular hyperbola problem requiring students to use the given tangent formula with two unknowns, set up simultaneous equations by substituting the intersection point, and solve for the parameters. While it involves multiple steps and algebraic manipulation, it follows a predictable pattern typical of FP1 conic sections questions with no novel insight required.
8. The rectangular hyperbola \(H\) has equation \(x y = c ^ { 2 }\), where \(c\) is a positive constant. The point \(P \left( c t , \frac { c } { t } \right) , t \neq 0\), is a general point on \(H\).
An equation for the tangent to \(H\) at \(P\) is given by
$$y = - \frac { 1 } { t ^ { 2 } } x + \frac { 2 c } { t }$$
The points \(A\) and \(B\) lie on \(H\).
The tangent to \(H\) at \(A\) and the tangent to \(H\) at \(B\) meet at the point \(\left( - \frac { 6 } { 7 } c , \frac { 12 } { 7 } c \right)\).
Find, in terms of \(c\), the coordinates of \(A\) and the coordinates of \(B\).
8. The rectangular hyperbola $H$ has equation $x y = c ^ { 2 }$, where $c$ is a positive constant. The point $P \left( c t , \frac { c } { t } \right) , t \neq 0$, is a general point on $H$.
An equation for the tangent to $H$ at $P$ is given by
$$y = - \frac { 1 } { t ^ { 2 } } x + \frac { 2 c } { t }$$
The points $A$ and $B$ lie on $H$.\\
The tangent to $H$ at $A$ and the tangent to $H$ at $B$ meet at the point $\left( - \frac { 6 } { 7 } c , \frac { 12 } { 7 } c \right)$.\\
Find, in terms of $c$, the coordinates of $A$ and the coordinates of $B$.\\
\hfill \mbox{\textit{Edexcel FP1 2014 Q8 [5]}}