When does the equation $ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0$ represent an ellipse?

For the curve $x^{2}+4xy+8y^{2}=64$, the tangents are parallel to the $x$-axis only at the points

$A$ chord through the point $(1,-2)$ cuts the curve $3x^2-y^2-2x+4y=0$ at $P$ and $Q$. If $PQ$ subtends an angle $\theta$ at the origin,then $\theta$ equals (in $^{\circ}$)

The equation $14x^2 - 4xy + 11y^2 - 44x - 58y + 71 = 0$ represents

Let $C$ be a curve $ax^2+2hxy+by^2+2gx+2fy+c=0$ in a Cartesian plane. By rotating the coordinate axes through an angle $\frac{\pi}{4}$ in the positive direction,if the transformed equation of $C$ is $Y^2+XY-X=0$,then $(h^2-ab)-2gf=$

If the chord through $(1, -2)$ cuts the curve $3x^2 - y^2 - 2x + 4y = 0$ at $P$ and $Q$,then the angle subtended by $PQ$ at the origin is (in $^{\circ}$)