The answer
(a) \((x+2)^2 + (y+2)^2 = 82\)
(b) \(-3 < \alpha < -1\)
O-Level A-Math 2025 Paper 1 Question 13 · Verified worked solution by the Genius Plus Academy teaching team
What this question tests
This is Question 13 of the O-Level A-Math 2025 Paper 1. It tests equation of a circle, in the Coordinate geometry (A-Math) area. It is worth 9 marks: 7 + 2. It is a worded / diagram-based question, so open your Ten-Year Series (TYS) or the official paper at this question, then follow our full worked solution below.
(a) The chord \(AB\), \(2y - 3x = 17\), has gradient \(\dfrac{3}{2}\). A diameter passes through the centre, so the diameter parallel to \(AB\) is the line through \(\left(-\tfrac12, \tfrac14\right)\) with gradient \(\tfrac32\): \[y - \tfrac14 = \tfrac32\left(x + \tfrac12\right).\] The centre lies on \(y = -2\), so substitute \(y = -2\): \[-2 - \tfrac14 = \tfrac32\left(x + \tfrac12\right) \;\Rightarrow\; -\tfrac94 = \tfrac32\left(x + \tfrac12\right) \;\Rightarrow\; x + \tfrac12 = -\tfrac32 \;\Rightarrow\; x = -2.\] So the centre is \((-2, -2)\). The radius is the distance to \(A(-1, 7)\): \[r^2 = (-1+2)^2 + (7+2)^2 = 1 + 81 = 82.\] The equation of the circle is \((x+2)^2 + (y+2)^2 = 82\).
(b) The point \((7, \alpha)\) is inside the circle when \((7+2)^2 + (\alpha+2)^2 < 82\), i.e. \(81 + (\alpha+2)^2 < 82\), so \((\alpha+2)^2 < 1\), giving \(-1 < \alpha + 2 < 1\), hence \(-3 < \alpha < -1\).
Answer: (a) \((x+2)^2 + (y+2)^2 = 82\)
(b) \(-3 < \alpha < -1\)
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 equation of a circle question from Coordinate geometry (A-Math), worth 9 marks: 7 + 2.
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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