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Question

A circle with centre O passes through the vertex A of an equilateral triangle ABC and touches BC at its midpoint M. The circle cuts AB at D and AC at E.

If x is the area of the circle and y is the area of the triangle such that \(z = \left(\frac{x}{y}\right)^2\), then which one of the following is correct?

This question was previously asked in
CDS 2 2026 Maths Question Paper (13-Sep-2026)
The correct answer is

1.5 < z < 2

The radius found above is \(r=h/2=\frac{\sqrt3}{4}a\), so \(x=\pi r^2=\frac{3\pi}{16}a^2\). The triangle's area is \(y=\frac{\sqrt3}{4}a^2\). Then \(\frac{x}{y}=\frac{\sqrt3\,\pi}{4}\approx1.360\), so \(z=\left(\frac{x}{y}\right)^2\approx1.85\), which lies strictly between 1.5 and 2.

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