The answer
\((1, 0)\)
O-Level A-Math 2024 Paper 1 Question 4 · Verified worked solution by the Genius Plus Academy teaching team
The question
The line \(2y + x = 1\) cuts the curve \(x^2 + y^2 - 2x = 4\) at \(A\) and \(B\). Find the coordinates of the midpoint of \(AB\). [6]
From the line, \(x = 1 - 2y\). Substitute into the curve: \[(1 - 2y)^2 + y^2 - 2(1 - 2y) = 4 \Rightarrow (1 - 4y + 4y^2) + y^2 - 2 + 4y = 4 \Rightarrow 5y^2 - 1 = 4,\] so \(5y^2 = 5\), \(y = \pm 1\). Then \(y = 1 \Rightarrow x = -1\) (point \(A(-1, 1)\)) and \(y = -1 \Rightarrow x = 3\) (point \(B(3, -1)\)). Midpoint \(= \left(\dfrac{-1 + 3}{2}, \dfrac{1 + (-1)}{2}\right) = (1, 0)\).
Answer: \((1, 0)\)
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 intersection of a line and a circle question from Coordinate geometry (A-Math), worth 6 marks.
Yes. IP (Integrated Programme) schools teach the same O-Level Mathematics content; they just sequence it differently and set their own internal exams, so these worked solutions apply to IP students too.
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