Jitender Gupta and Divya Malik Solutions for Exercise 3: CHALLENGERS

If $G$ be the centroid of a $△ABC$ and $P$ be any other point in the plane, then $G{A}^2+G{B}^2+G{C}^2+3G{P}^2$ is equal to

A figure is shown below

If we rotate this graph about $O$ at an angle of $180°$ in anticlockwise direction, then the intersection point of diagonals will be

If $\left(-4,3\right)$ and$\left(4,3\right)$ are two vertices, of an equilateral triangle and the origin lies in the interior of the triangle, then the coordinates of the third vertex will be:

Suppose there are four points $A\left(2,4\right),B\left(6,4\right),C\left(6,6\right)\text{ and} D\left(2,6\right)$  which lie in the first quadrant. If we rotate only the axes at an angle of $90°$ in anticlockwise direction, then what will be the new coordinates of the point $C$ and what will be the name of the figure, when we join adjacent points.

 Suppose $PQ$ be a pole, whose coordinates are $P\left(1,3\right)$ and $Q\left(3,3\right)$ and $A$ be the position of a man whose coordinates are $\left(1,1\right)$. If a pole makes an angle of elevation to the point $A$, then the angle $\theta$ is

 Suppose $PQ$ be a pole, whose coordinates are $P\left(1,3\right)$ and $Q\left(3,3\right)$ and $A$ be the position of a man whose coordinates are $\left(1,1\right)$. If we shift the origin at $\left(1,1\right)$, then the angle $\theta$ is?

The sum of the squares of the distances of a moving point $\left(x,y\right)$ from two fixed points $\left(a,0\right)$ and$\left(-a,0\right)$ is equal to a constant quantity $2b^2$. The value of  $x^2+y^2+a^2$ is equal to 

A point$\left(x,y\right)$ moves, so that the sum of its distances from $\left(ae,0\right)$ and$\left(-ae,0\right)$ is $2a$. If $b^2=a^2\left(1-e^2\right)$, then $\frac{x^2}{a^2}+\frac{y^2}{b^2}$ is equal to