Relative Position of a Point with Respect to a Line

Q.

There are 16 points in a plane out of which 6 are collinear, then how many lines can be drawn by joining these points

1. 55

2. 106

3. 105

4. 60

Q. If $(a,a^2)$ falls inside the angle made by the lines $y=\frac{x}{2},x>0,$ and $y=3x,x>0,$ then a belongs to

1. $(0,\frac{1}{2})$
2. $(3,\infty )$
3. $(\frac{1}{2},3)$
4. $(-3,\frac{1}{2})$

Q. Find the range of $\theta$ in the interval $(0,\pi )$ such that the point $(3,5)$ and $(\sin \theta ,\cos \theta )$ lie on the same side of the line $x+y–1=0.$

Q. For points $P=(x_1,y_1)\text{and}Q=(x_2,y_2)$ of the co-ordinate plane, a new distance $d(P,Q)$ is defined by $d(P,Q)=|x_1-x_2|+|y_1-y_2|$. Let $O=(0,0)$ and $A=(3,2)$. Prove that the set of points in the first quadrant which are equidistant (with respect to the new distance) from $O$ and $A$ consinsts of the union of a line segment of finite length and an infinite ray. Sketch this set in a labelled diagram.