Parametric Form of a Straight Line

Q.

If $\left(\sin \theta ,\cos \theta \right)$ and $\left(3,2\right)$ lie on the same side of the line $x+y=1$, then $\theta$ lies between

1. $\left(0,\frac{\pi }{2}\right)$

2. $\left(0,\pi \right)$

3. $\left(\frac{\pi }{4},\frac{\pi }{2}\right)$

4. $\left(0,\frac{\pi }{4}\right)$

Q. The parallelism condition for two straight lines one of which is specified by the equation ax+by+c=0 the other being represented parametrically by $x=\alpha t+\beta ,y=\gamma t+\delta$ is given by

1. $\alpha \gamma -b\alpha =0,\beta =\delta =c=0$
2. $a\alpha -b\gamma =0,\beta =\delta =0$
3. $a\alpha +b\gamma =0$
4. $a\gamma =b\alpha =0$

Q.

The middle point of the line segment joining (3, -1)and (1, 1) is shifted by two units (in the sense of increasing y) Perpendicular to the line segment. Then, the coordinates of the point in the new position are

1. $(2-\sqrt{2},2)$

2. $(2,2-\sqrt{3})$

3. $(2+\sqrt{2},\sqrt{2})$

4. $(2+2,\sqrt{3})$

Q. In what direction a line be drawn through the point (1, 2) so that its points of intersection with the line x+y=4 is at a distance $\frac{\sqrt{6}}{3}$ from the given point

1. ${30}^{\circ }$
2. ${45}^{\circ }$
3. ${60}^{\circ }$
4. ${75}^{\circ }$

Q.

The middle point of the line segment joining (3, -1)and (1, 1) is shifted by two units (in the sense of increasing y) Perpendicular to the line segment. Then, the coordinates of the point in the new position are

1. $(2-\sqrt{2},2)$

2. $(2,2-\sqrt{3})$

3. $(2+\sqrt{2},\sqrt{2})$

4. $(2+2,\sqrt{3})$

Q. A line passing through $A(-5,-4)$ meets the line $x+3y+2=0,2x+y+4=0$ and $x-5-y=0$ at $B,C$ and $D$ respectively. If ${\left(\frac{15}{AB}\right)}^2+{\left(\frac{10}{AC}\right)}^2={\left(\frac{6}{AD}\right)}^2$, then the equation of line is

1. $2x+3y+22=0$
2. $5x+4y+7=0$
3. $3x-2y+3=0$
4. None of these