2025 NJC P2 Q2
2025 NJC P2 Q2
11 marks
The curve \(E\) has equation
\[(x-a)^2 + 4y^2 = 4, \text{ where}\hspace{0.5em} 0 < a < 2.\]
- Sketch \(E\).[2]
You are now given that \(a=1\).
- Show that the equation of a normal to \(E\) at \(x=0\) is \(y=2\sqrt{3}x-\frac{\sqrt{3}}{2}\).[4]
- The region \(R\) is bounded by \(E\), the normal in part (ii) and the \(y\)-axis. Find the volume of the solid generated when \(R\) is rotated through \(2\pi\) radians about the \(y\)-axis.[5]
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Answer:(i) Ellipse with centre \((a,0)\), horizontal semi-axis \(2\), vertical semi-axis \(1\), and vertices \((a\pm2,0)\), \((a,\pm1)\) (ii) \(y=2\sqrt3x-\dfrac{\sqrt3}{2}\) (iii) \(0.517\text{ units}^3\)
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