If $x^2-y^2+2 h x y+2 g x+2 f y+c=0$ is the locus of a point, which moves such that it is always equidistant from the lines $x+2 y+7=0$ and $2 x-y$ $+8=0$, then the value of $\mathrm{g}+\mathrm{c}+\mathrm{h}-\mathrm{f}$ equals

Let the points of intersections of the lines $x-y+1=0$, $x-2 y+3=0$ and $2 x-5 y+11=0$ are the mid points of the sides of a triangle $A B C$. Then the area of the triangle $\mathrm{ABC}$ is .... .

The area of the parallelogram formed by the lines $y = mx,\,y = mx + 1,\,y = nx$ and $y = nx + 1$ equals

In the triangle $ABC$ with vertices $A$$(2,3), B(4,-1)$ and $C(1,2),$ find the equation and length of altitude from the vertex $A$.

The base of an equilateral triangle with side $2 a$ lies along the $y$ -axis such that the mid-point of the base is at the origin. Find vertices of the triangle.

The opposite vertices of a square are $(1, 2)$ and $(3, 8)$, then the equation of a diagonal of the square passing through the point $(1, 2)$, is