The answer
(a) \(C(-5, 12)\), radius \(13\)
(b) \((0, 0)\) and \((0, 24)\)
(c) \((-20, 48)\) or \((10, -24)\)
O-Level A-Math 2023 Paper 1 Question 10 · Verified worked solution by the Genius Plus Academy teaching team
The question
\(x^2 + y^2 + 10x - 24y = 0\), centre \(C\).
(a) Find \(C\) and the radius. [4]
(b) Find where the circle meets the \(y\)-axis. [2] \(X\) is on line \(OC\) with \(CX = 3\,OC\) (\(O\) the origin).
(c) Find the possible coordinates of \(X\). [4]
(a) Completing the square: \((x + 5)^2 - 25 + (y - 12)^2 - 144 = 0 \Rightarrow (x + 5)^2 + (y - 12)^2 = 169\). So \(C = (-5, 12)\), radius \(= 13\).
(b) At \(x = 0\): \(y^2 - 24y = 0 \Rightarrow y(y - 24) = 0 \Rightarrow y = 0\) or \(24\). Points \((0, 0)\) and \((0, 24)\).
(c) Points on line \(OC\) are \(P(t) = t(-5, 12)\) (\(t = 0\) at \(O\), \(t = 1\) at \(C\)). The distance from \(C\) is \(|t - 1|\cdot|OC| = 13|t - 1|\), and \(|OC| = 13\). Setting \(13|t - 1| = 3(13)\) gives \(|t - 1| = 3\), so \(t = 4\) or \(t = -2\). Then \(X = 4(-5, 12) = (-20, 48)\) or \(X = -2(-5, 12) = (10, -24)\).
Answer: (a) \(C(-5, 12)\), radius \(13\)
(b) \((0, 0)\) and \((0, 24)\)
(c) \((-20, 48)\) or \((10, -24)\)
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 (complete the square) question from Coordinate geometry (A-Math), worth 10 marks: 4 + 2 + 4.
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