The answer
(a) shown
(b) \((-2a, 3a)\) and \((-2a, -a)\)
(c) verified
(d) \((-5a, 0)\)
O-Level A-Math 2021 Paper 2 Question 9 · Verified worked solution by the Genius Plus Academy teaching team
What this question tests
This is Question 9 of the O-Level A-Math 2021 Paper 2. It tests equation of a circle, in the Coordinate geometry (A-Math) area. It is worth 12 marks: 3 + 2 + 1 + 6. 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) With \(k = 4\), the circle has centre \((-2a, a)\) and radius \(\sqrt{4a^2} = 2a\). The perpendicular distance from the centre to the \(y\)-axis (\(x = 0\)) is \(|-2a| = 2a\), equal to the radius, so the \(y\)-axis touches the circle: it is a tangent.
(b) A tangent parallel to the \(x\)-axis touches at the top and bottom of the circle, where \(x = -2a\) (the centre's \(x\)) and \(y = a \pm 2a\). So the points are \((-2a, 3a)\) and \((-2a, -a)\).
(c) With \(k = 5\), the equation is \((x + 2a)^2 + (y - a)^2 = 5a^2\). At \(O(0, 0)\): \((2a)^2 + (-a)^2 = 4a^2 + a^2 = 5a^2 = \) RHS, so the circle passes through \(O\). (verified)
(d) The centre \(C(-2a, a)\) is the midpoint of diameter \(OP\), so \(P = (2(-2a) - 0,\ 2(a) - 0) = (-4a, 2a)\). The gradient of \(OP\) is \(\dfrac{2a}{-4a} = -\tfrac12\); the tangent at \(P\) is perpendicular to \(OP\), with gradient \(2\). Its equation is \(y - 2a = 2(x + 4a) \Rightarrow y = 2x + 10a\). Setting \(y = 0\) gives \(x = -5a\), so the tangent meets the \(x\)-axis at \((-5a, 0)\).
Answer: (a) shown
(b) \((-2a, 3a)\) and \((-2a, -a)\)
(c) verified
(d) \((-5a, 0)\)
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 equation of a circle question from Coordinate geometry (A-Math), worth 12 marks: 3 + 2 + 1 + 6.
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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