Consider an elliptic disc having uniform distribution of mass M as shown in figure. The moment of inertia (M.I.) about the major and minor
axis are $\frac{1}{4} {mb}^{2} and \frac{1}{4} {ma}^{2}$ respectively. Then

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The M.I. about a line, in its plane, passing through its centre and inclined at an angle $θ$ to the major axis is $\frac{1}{4} m (b^{2} \cos^{2} θ + a^{2} \sin^{2} θ)$

The M.I. about a chord of length 2r passing through O and lying in plane of elliptic disc is $\frac{1}{4} \frac{{Ma}^{2} b^{2}}{r^{2}}$

The M.I. about a line perpendicular to the plane of elliptic disc and passing through the centre O is $\frac{1}{4} M (a^{2} + b^{2})$

The M.I. about a line, in its plane, passing through its centre and inclined at an angle $θ$ to the minor axis is $\frac{1}{4} m (b^{2} \sin^{2} θ + a^{2} \cos^{2} θ)$

answer is A, B, C.

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Detailed Solution

We know that
$\begin{array}{l} I_{PQ} = I_{x} \cos^{2} θ + I_{y} \sin^{2} θ - I_{xy} \sin 2 θ \\ I_{PQ} = \frac{1}{4} {mb}^{2} \cos^{2} θ + \frac{1}{4} {ma}^{2} \sin^{2} θ [∵ I_{xy} = 0] \end{array}$
The equation of ellipse is
$\begin{array}{l} \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 \\ \frac{r^{2}}{a^{2}} \cos^{2} θ + \frac{r^{2}}{b^{2}} \sin^{2} θ = 1 \\ a^{2} \sin^{2} θ + b^{2} \cos^{2} θ = \frac{a^{2} b^{2}}{r^{2}} \\ ∴ I_{PQ} = \frac{1}{4} m (b^{2} \cos^{2} θ + a^{2} \sin^{2} θ) = \frac{1}{4} m \frac{a^{2} b^{2}}{r^{2}} \end{array}$