The answer
(a) \(-\dfrac{10}{h + 7}\)
(b) \(h = 9\) or \(h = -2\)
O-Level A-Math 2023 Paper 1 Question 13 · Verified worked solution by the Genius Plus Academy teaching team
The question
The perpendicular bisector of \(A(2, h)\) and \(B(-8, -7)\) passes through \(X\left(h, -\dfrac{13}{2}\right)\).
(a) Express the gradient of the perpendicular bisector in terms of \(h\). [3]
(b) Hence find the possible values of \(h\). [7]
(a) Gradient of \(AB = \dfrac{h - (-7)}{2 - (-8)} = \dfrac{h + 7}{10}\). The perpendicular bisector has gradient \(-\dfrac{10}{h + 7}\).
(b) The perpendicular bisector passes through the midpoint \(M = \left(-3, \dfrac{h - 7}{2}\right)\) and through \(X\left(h, -\dfrac{13}{2}\right)\), so the gradient of \(MX\) equals the gradient in (a): \[\frac{\frac{h - 7}{2} - \left(-\frac{13}{2}\right)}{-3 - h} = -\frac{10}{h + 7} \Rightarrow \frac{\frac{h + 6}{2}}{-(h + 3)} = -\frac{10}{h + 7} \Rightarrow \frac{h + 6}{2(h + 3)} = \frac{10}{h + 7}.\] Cross-multiplying: \((h + 6)(h + 7) = 20(h + 3) \Rightarrow h^2 + 13h + 42 = 20h + 60 \Rightarrow h^2 - 7h - 18 = 0 \Rightarrow (h - 9)(h + 2) = 0\). So \(h = 9\) or \(h = -2\).
Answer: (a) \(-\dfrac{10}{h + 7}\)
(b) \(h = 9\) or \(h = -2\)
Same structure, different numbers
Swap the constants, dress a quadratic as a length, hide a derivative inside an integral, and a student sees a brand new problem. The structure underneath is the same, and so is the method. Once a student can name the structure, a whole row of questions that look different start to open the same way.
That is where marks really leak: in choosing the method, not in the algebra that follows. We call it Lock and Key, name the lock, then the key follows.
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Genius Plus Academy · O-Level & IP Mathematics
Our O-Level A-Math tuition trains the same recognise-the-structure method these worked solutions show, taught by a team that has marked these papers for years. It runs within our weekly Secondary Math programme, Sec 1 to 4 and IP.
It is a perpendicular bisector question from Coordinate geometry (A-Math), worth 10 marks: 3 + 7.
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