Let the line $x+y=1$ meet the axes of $x$ and $y$ at $A$ and $B$, respectively. A right angled triangle $A M N$ is inscribed in the triangle $O A B$, where $O$ is the origin and the points $M$ and $N$ lie on the lines $OB$ and $A B$, respectively. If the area of the triangle AMN is $\frac{4}{9}$ of the area of the triangle $OAB$ and $AN : NB =\lambda: 1$, then the sum of all possible value$(s)$ of is $\lambda$ :

If the vertices $P$ and $Q$ of a triangle $PQR$ are given by $(2, 5)$ and $(4, -11)$ respectively, and the point $R$ moves along the line $N: 9x + 7y + 4 = 0$, then the locus of the centroid of the triangle $PQR$ is a straight line parallel to

The diagonals of a parallelogram $PQRS$ are along the lines $x + 3y = 4$ and $6x - 2y = 7$. Then $PQRS$ must be a

Two lines are drawn through $(3, 4)$, each of which makes angle of $45^\circ$ with the line $x - y = 2$, then area of the triangle formed by these lines is

If one vertex of an equilateral triangle of side $'a'$ lies at the origin and the other lies on the line $x - \sqrt{3} y = 0$ then the co-ordinates of the third vertex are :

If a variable line drawn through the point of intersection of straight lines $\frac{x}{\alpha } + \frac{y}{\beta } = 1$and $\frac{x}{\beta } + \frac{y}{\alpha } = 1$ meets the coordinate axes in $A$ and $B$, then the locus of the mid point of $AB$ is