Two equal sides of an isosceles triangle are along $ -x + 2y = 4 $ and $ x + y = 4 $. If $ m $ is the slope of its third side, then the sum, of all possible distinct values of $ m $, is:

If A and B are the points of intersection of the circle $x^2 + y^2 - 8x = 0$ and the hyperbola $\frac{x^2}{9} - \frac{y^2}{4} = 1$ and a point P moves on the line $2x - 3y + 4 = 0$, then the centroid of $\Delta PAB$ lies on the line :

Let the points $\left(\frac{11}{2}, \alpha\right)$ lie on or inside the triangle with sides $x+y=11, x+2 y=16$ and $2 x+3 y=29$. Then the product of the smallest and the largest values of $\alpha$ is equal to :

Let the lines $3 x-4 y-\alpha=0,8 x-11 y-33=0$, and $2 x-3 y+\lambda=0$ be concurrent. If the image of the point $(1,2)$ in the line $2 x-3 y+\lambda=0$ is $\left(\frac{57}{13}, \frac{-40}{13}\right)$, then $|\alpha \lambda|$ is equal to