Dr. SK Goyal Solutions for Chapter: Coordinate Systems and Coordinates, Exercise 7: Multiple Choice I

If $\left(3,-4\right)$ and $\left(-6,5\right)$ are the extremities of a diagonal of a parallelogram and $\left(-2,1\right)$ is the third vertex, then its fourth vertex is-

The medians of a triangle meet at $(0,-3)$ and two vertices are at $(-1,4)$ and $(5,2)$. Then the third vertex is :

The sides of a triangle are $3x+4y, 4x+3y$ and $5x+5y$ units, where $x>0, y>0$. The triangle is :

A triangle $ABC$ which is right angled at $A$ has points $A$ and $B$ at $\left(2,3\right)$ and $\left(0,-1\right)$ respectively. If $BC=5$ units, then the point $C$ is

$ABCD$ is a square whose vertices $A,B,C$ and $D$ are $\left(0,0\right),\left(2,0\right),\left(2,2\right)$ and $\left(0,2\right)$ respectively. This square is rotated in the $X-Y$ plane with an angle of $30°$ in anticlockwise direction about an axis passing through the vertex $A$ the equation of the diagonal $BD$ of this rotated square is......... If $E$ is the centre of the square, the equation of the circumcircle of the triangle $ABE$ is :

If $t_1,t_2$ and $t_3$ are distinct, the points $\left(t_1,2a{t}_1+a{t}_1^3\right), \left(t_2,2a{t}_2+a{t}_2^3\right)$ and $\left(t_3,2a{t}_3+a{t}_3^3\right)$ are collinear, if :

Let $d\left(P,OA\right)\le \min \left\{d\left(P,AB\right),d\left(P,BC\right),d\left(P,OC\right)\right\}$ where $d$ denotes the distance from the point to the corresponding line and $S$ be the region consisting of all those points $P$ inside the rectangle $OABC$ such that $O=\left(0,0\right), A=\left(3,0\right), B=\left(3,2\right)$ and $C=\left(0,2\right)$, which satisfy the above relation, then area of the region $S$ is:

Statement (i): The points $\left(-1,0\right),\left(4,-2\right)$ and $\left(\cos 2\theta ,\sin 2\theta \right)$ are collinear

Statement (ii): The points $\left(-1,0\right),\left(4,-2\right)$ and $\left(\frac{1-{\tan }^2\theta }{1+{\tan }^2\theta },\frac{2\tan \theta }{1+{\tan }^2\theta }\right)$ are collinear