By rotating the axes about the origin in anti-clockwise direction with certain angle, if the equation $x^2+4 x y+y^2=1$ is transformed to $\frac{x^2}{a^2}-\frac{y^2}{b^2}=l$, then $\sqrt{\frac{a^2+b^2}{a^2}}=$

If a tangent to the hyperbola $x y=-1$ is also a tangent to the parabola $y^2=8 x$, then the equation of that tangent is

The distance between the tangents of the hyperbola $2 x^2-3 y^2=6$ which are perpendicular to the line $x-2 y+5=0$ is

The tangents drawn to the hyperbola $5 x^2-9 y^2=90$ through a variable point $P$ make the angles $\alpha$ and $\beta$ with its transverse axis. If $\alpha, \beta$ are the complementary angles then the locus of $P$ is