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2024 NYJC H1 Q3

2024 NYJC H1 Q3

6 marks

The curve \(C\) has equation \(y = \sqrt{k - x}\) , where \(k\) is a constant. The line \(L\) has equation \(y + x = 2k\) .

  1. Given that \(C\) and \(L\) intersect, find the range of possible values of \(k\).[3]

Let \(k = \frac{1}{4}\).

  1. Sketch the graphs of \(C\) and \(L\) on a single diagram, stating the coordinates of any intersections with the axes.[3]

Solution:

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Answer:(a) \(k\le\dfrac14\). (b) \(C\): \(\left(\dfrac14,0\right),\left(0,\dfrac12\right)\); \(L\): \(\left(\dfrac12,0\right),\left(0,\dfrac12\right)\), with common tangency at \(\left(0,\dfrac12\right)\).

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