Parametric Form of a Straight Line

Q. If the distance of the point $(2,3)$ from the line $2x-3y+9=0$ measured along a line $x-y+1=0$ is $k$ units, then the value of $k^2$ is

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. 

3. 

4. 

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.

For all values of $\theta$, the lines represented by the equation
$(2cos\theta +3sin\theta )x+(3cos\theta -5sin\theta )y-(5cos\theta -2sin\theta )=0$

1. pass through a fixed point

2. pass through the point (1, 1)

3. pass through a fixed point whose reflection in the line
   $x+y=\sqrt{2}$ is$(\sqrt{2}-1,\sqrt{2}-1)$

4. pass through the origin

Q. The slope of the straight line passing through A(3, 2) is $\frac{3}{4}$. The coordinates of the points that are 5 units away from A are:

1. (7, 5)
2. (-1, -1)
3. (-7, -5)
4. (1, 1)

Q. A variable line $L$ is drawn through $O(0,0)$ to meet the lines $L_1:x+2y-3=0$ and $L_2:x+2y+4=0$ at points $M$ and $N$ respectively. A point $P$ is taken on line $L$ such that $\frac{1}{O{P}^2}=\frac{1}{O{M}^2}+\frac{1}{O{N}^2}$. Then the locus of $P$ is

1. $x^2+4y^2=\frac{144}{25}$
2. $(x+2y{)}^2=\frac{144}{25}$
3. $4x^2+y^2=\frac{144}{25}$
4. $(x-2y{)}^2=\frac{144}{25}$

Q. The equation of the medians of a triangle formed by the lines $x+y-6=0,x-3y-2=0$ and $5x-3y+2=0$, is

1. $x=2,x-9y-14=0$ and $7x-9y+2=0$
2. $x=2,x+9y-14=0$ and $7x-9y-2=0$
3. $x=2,x-9y-14=0$ and $7x+9y+2=0$
4. $x=2,x+9y+14=0$ and $x-9y-2=0$

Q. The slope of a straight line passing through $A(-2,3)$ is $-\frac{4}{3}$. The point(s) on the line that are $10$ units away from $A$ is/are

1. $(8,0)$
2. $(-8,11)$
3. $(4,-5)$
4. $(4,11)$

Q. A line which passes through $P(4,5)$ and making an angle of ${30}^{\circ }$ with positive direction of $x-$axis. Then coordinates of point which is at a distance $4$ unit's from the line on either side of $P$, is

1. $(4+2\sqrt{3},7),(4-2\sqrt{3},3)$
2. $(4-2\sqrt{3},7),(4+2\sqrt{3},3)$
3. $(4-2\sqrt{3},7),(4+2\sqrt{3},7)$
4. $(4+2\sqrt{3},3),(4+2\sqrt{3},7)$

Q. Parametric form of a straight line passing through $(4,5)$ and making an angle ${60}^{\circ }$ with $x$-axis in the positive direction is
(where $\lambda$ be any parameter)

1. $x=\lambda +4,y=\sqrt{3}\lambda +5$
2. $x=\sqrt{3}\lambda -4,y=\lambda +5$
3. $x=\sqrt{3}\lambda +4,y=\lambda +5$
4. $x=\lambda -4,y=\lambda +5$