Hey Rui, can you please help me with Q15a) i and iii? -I don't understand the solutions provided
For i) why is p a double root if the hyperbola touches the ellipse at P?
For iii) what do they mean by the parameter at the point Q? Why is it -p? And how do they know that O is the midpoint of PQ?
Thank you in advance(:
Because \(P\) is the point \( \left(cp, \frac{c}{p} \right) \), the parameter at \(P\) is \(p\). That's literally it - the parameter at a point is just the parameter that represents the coordinates of that point. The reason why \(Q\) must therefore be \(-p\) is because clearly the ellipse and the hyperbola intersect at only two distinct points. So because \( \left( cp, \frac{c}{p} \right) \) corresponds to \(P\), this leaves us with \( \left( -cp, -\frac{c}{p} \right) \) corresponding to \(Q\). So clearly \(-p\) must be the parameter representing the point \(Q\).
(It's the same as the parabola \(x^2 = 4ay\). If a point is marked \( P(2ap, ap^2)\), then the parameter at \(P\) is just \(p\).)
It is then easy to show by the midpoint formula that the midpoint of \(P\) and \(Q\) is the origin.
For example, the quadratic equation \( x^2 - 2x + 1 = 0\) has only one unique solution. (In particular, for that one it will be \(x = 1\).) That solution is also a double root of the equation.
It's quite abstract to argue formally, but essentially the idea is to generalise the intersection between a conic and a tangent line at the point of contact, to the intersection between two "touching" conics at their point of contact instead.