Q.The equation of the ellipse having eccentricity $\dfrac{\sqrt3}{2}$ and passing through $\left(-\sqrt8, \sqrt3\right)$ is
A) $4x^2 + y^2 = 4$ B) $x^2 + 4y^2 = 100$ C) $4x^2 + y^2 = 100$ D) $x^2 + 4y^2 = 4$

For eccentricity $\dfrac{\sqrt3}{2}$: $e^2=\dfrac34=1-\dfrac{b^2}{a^2} \Rightarrow \dfrac{b^2}{a^2}=\dfrac14$, i.e. the ellipse's two axis-denominators are always in a $4:1$ ratio. Every one of the four printed options ($4x^2+y^2=4$, $x^2+4y^2=100$, $4x^2+y^2=100$, $x^2+4y^2=4$) already has exactly this $4{:}1$ ratio, so eccentricity alone cannot distinguish between them — only the point of the curve can, via the constant on the right-hand side.

Substituting the point $\left(-\sqrt8,\sqrt3\right)$, i.e. $x^2=8,\,y^2=3$, into each candidate family: $4x^2+y^2=4(8)+3=35$ and $x^2+4y^2=8+4(3)=20$ — neither equals the constant ($4$ or $100$) printed in any of the four options. …

First check what constraint the eccentricity alone imposes (here, only the axis ratio, not the scale); only then bring in the point — and if neither is …

Silently picking the 'closest-looking' option without flagging that the numbers don't actually check out — this is the …