If two vertices of an equilateral triangle are $(1, 0)$ and $(3, 0)$,and the third vertex lies in the first quadrant,find the area of the triangle.

Two consecutive sides of a parallelogram are $4x + 5y = 0$ and $7x + 2y = 0.$ If the equation to one diagonal is $11x + 7y = 9,$ then the equation of the other diagonal is

The bisectors of the angles of a parallelogram enclose a

The intersection of three lines $x-y=0$,$x+2y=3$,and $2x+y=6$ forms a:

The vertices of a triangle are $(at_1t_2, a(t_1 + t_2))$,$(at_2t_3, a(t_2 + t_3))$,and $(at_3t_1, a(t_3 + t_1))$. Find the coordinates of its orthocentre.

Let the line $x+y=1$ meet the axes of $x$ and $y$ at $A$ and $B$,respectively. $A$ right-angled triangle $AMN$ is inscribed in the triangle $OAB$,where $O$ is the origin and the points $M$ and $N$ lie on the lines $OB$ and $AB$,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 values of $\lambda$ is: