Let ${ }^n C_{r-1}=28,{ }^n C_r=56$ and ${ }^n C_{r+1}=70$. Let $A (4 \cos t, 4 \sin t ), B (2 \sin t ,-2 \cos t )$ and $C \left(3 r - n , r ^2- n -1\right)$ be the vertices of a triangle $A B C$, where $t$ is a parameter. If $(3 x-1)^2+(3 y)^2=\alpha$, is the locus of the centroid of triangle $ABC$ , then $\alpha$ equals :

$A(-1, 1)$, $B(5, 3)$ are opposite vertices of a square in $xy$-plane. The equation of the other diagonal (not passing through $(A, B)$ of the square is given by

Two mutually perpendicular straight lines through the origin from an isosceles triangle with the line $2x + y = 5$ . Then the area of the triangle is :

One vertex of the equilateral triangle with centroid at the origin and one side as $x + y - 2 = 0$ is

The base $BC$ of a triangle $ABC$ is bisected at the point $(p, q)$ and the equation to the side $AB \,\,ane\,\, AC$ are $px + qy = 1 \,\,ane\,\, qx + py = 1$ . The equation of the median through $A$ is :

A vertex of equilateral triangle is $(2, 3)$ and equation of opposite side is $x + y = 2,$ then the equation of one side from rest two, is