What is the set of values of a for which the point ( $2 a, a+1$ ) is an interior point of the larger segment of the circle $x^2+y^2-2 x-2 y-8=0$ made by the chord $x-y+1=0$.

Tangents are drawn to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ at points where it is intersected by the line $l x+m y+n=0$. The point of intersection of tangents at these points is

The locus of the mid-point of the chords of the circle $x^2+y^2=16$ which are tangents to the hyperbola $9 x^2-16 y^2=144$ is $\left(x^2+y^2\right)^2=k x^2-l y^2$. Then, the sum of values of $k$ and $l$ is

If $A+B=\frac{\pi}{4}$, then $(1+\tan A)(1+\tan B)$ is equal to