Year: 2023
Paper: 3
Question Number: 1
Course: UFM Pure
Section: Conic sections
No solution available for this problem.
The total entry was a marginal increase on that of 2022 (by just over 1%). Two questions were attempted by more than 90% of candidates, another two by 80%, and another two by about two thirds. The least popular questions were attempted by more than a sixth of candidates. All the questions were perfectly answered by at least three candidates (but mostly more than this), with one being perfectly answered by eighty candidates. Very nearly 90% of candidates attempted no more than 7 questions. One general comment regarding all the questions is that candidates need to make sure that they read the question carefully, paying particular attention to command words such as "hence" and "show that".
Difficulty Rating: 1500.0
Difficulty Comparisons: 0
Banger Rating: 1500.0
Banger Comparisons: 0
The distinct points $P(2ap,\, ap^2)$ and $Q(2aq,\, aq^2)$ lie on the curve $x^2 = 4ay$, where $a > 0$.
\begin{questionparts}
\item Given that
\[(p+q)^2 = p^2q^2 + 6pq + 5\,,\tag{$*$}\]
show that the line through $P$ and $Q$ is a tangent to the circle with centre $(0,\, 3a)$ and radius $2a$.
\item Show that, for any given value of $p$ with $p^2 \neq 1$, there are two distinct real values of $q$ that satisfy equation $(*)$.
Let these values be $q_1$ and $q_2$. Find expressions, in terms of $p$, for $q_1 + q_2$ and $q_1 q_2$.
\item Show that, for any given value of $p$ with $p^2 \neq 1$, there is a triangle with one vertex at $P$ such that all three vertices lie on the curve $x^2 = 4ay$ and all three sides are tangents to the circle with centre $(0,\, 3a)$ and radius $2a$.
\end{questionparts}
Although this was a very popular question, being attempted by nearly 93% of the candidature, it was very narrowly beaten into second place by question 5. It was the third most successfully attempted with a mean score of approximately 9.5/20. Most candidates found the equation of the line, and of the circle and then solved simultaneously in part (i) to find common points, rather than using the perpendicular distance from a line formula; some using the distance formula misquoted it, with a common error being failure to include the modulus signs. Then they generally applied use of the discriminant, but with varying success. In part (ii), most candidates successfully expressed the given expression as a quadratic in q, obtained the determinant and the two required expressions using Vieta's formulas, but failed to fully demonstrate the inequality. Attempts at part (iii) were frequently inelegant and involved repeating work from previous parts of the question, rather than using the results of part (i) and (ii).