Distance between Parallel Lines

Q.

The plane passing through the points $(1,2,1),(2,1,2)$ and parallel to the line, $2x=3y,z=1$ also passes through the point

1. $(0,-6,2)$

2. $(0,6,-2)$

3. $(-2,0,1)$

4. $(2,0,-1)$

Q.

Three lines, $L_1:\overrightarrow{r}=\lambda \hat{i},\lambda \in R$, $L_2:\overrightarrow{r}=\hat{k}+\mu \hat{j},\mu \in R$ and $L_3:\overrightarrow{r}=\hat{i}+\hat{j}+\nu \hat{k},\nu \in R$ are given, for which point(s) $Q$ and $L_2$ can we find a point $P$ on $L_1$ and a point $R$ on $L_3$, so that $P,Q$ and $R$ are collinear?

1. $\hat{k}+\hat{j}$

2. $\hat{k}$

3. $\hat{k}+\frac{1}{2}\hat{j}$

4. $\hat{k}-\frac{1}{2}\hat{j}$

Q.

Distance between two lines represented by the pair of straight lines $x^2-6xy+9y^2+3x-9y-4=0$ is

1. $\frac{1}{2}$

2. $\frac{5}{\sqrt{2}}$

3. $\sqrt{10}$

4. $\sqrt{\frac{5}{2}}$

Q.

The equation of second degree$x^2+2\sqrt{2}xy+2y^2+4x+4\sqrt{2}y+1=0$ represents a pair of straight lines. The distance between them is

1. $2\sqrt[]{3}$

2. $2\sqrt[]{5}$

3. $2$

4. $10$

Q.

The equation of a line passing through the point (-3, 2, - 4) and equally inclined to the axes, are

1. 
   

2. 
   

3. 
   

4. None of these

Q. The point of intersection of the lines $\frac{x+1}{3}=\frac{y+3}{5}=\frac{z+5}{7}and\frac{x-2}{1}=\frac{y-4}{3}=\frac{z-6}{5}$ is

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

Q.

Equation of the line on which the length of the perpendicular from origin is 5 and the angle which this perpendicular makes with the x axis is ${60}^{\circ }$.

1. 

2. 

3. 

4. 

Q.

Transforming to parallel axes through a point $(p,q),$ the equation $2x^2+3xy+4y^2+x+18y+25=0$ becomes $2x^2+3xy+4y^2=1.$ Then

1. $p=-2,q=3$
2. $p=2,q=-3$
3. $p=3,q=-4$
4. $p=-4,q=3$

Q. Area enclosed by the common tangents of circle $x^2+y^2=4$ and ellipse $3x^2+5y^2=15$ is

1. $4$
2. $16$
3. $8$
4. $32$

Q.

The pairs of straight lines $x^2-3xy+2y^2=0$ and $x^2-3xy+2y^2+x-2=0$ form a

1. Square but not rhombus

2. Rhombus

3. Parallelogram

4. Rectangle but not a square

Q.

Prove that the lines $2x+3y=19$ and $2x+3y+7=0$ are equidistant from the line $2x+3y=6$

Q.

Determine whether the given planes are parallel or perpendicular and in case they are neither, find the angle between them.

2x- 2y+ 4z+ 5= 0 and 3x- 3y+ 6z- 1= 0

Q. A line through $(0,-2)$ and is perpendicular to the line $5x-y+7=0$. Then the equation of the line parallel to it and at a distance of $3$ units from it is/are

1. $x+5y+10-3\sqrt{26}=0$
2. $x+5y+10-\sqrt{13}=0$
3. $x+5y+10+3\sqrt{26}=0$
4. $x+5y+10+\sqrt{13}=0$

Q. The angle between the tangents drawn from the point $(4,\sqrt{2})$ to the ellipse $5x^2+4y^2=40.$

1. $\frac{\pi }{6}$
2. $\frac{\pi }{4}$
3. $\frac{\pi }{3}$
4. $\frac{\pi }{2}$

Q.

Find the equation of a line which is equidistant from the lines x = - 2 and x = 6.

Q.

Find the equation of a line equidistant from the lines y = 10 and y = - 2.

Q.

If line $y-x+2=0$ is shifted parallel to itself towards the $X$-axis by a perpendicular distance of $3\sqrt{2}\mathrm{units}$, then the equation of the new line may be:

1. $y=x+4$

2. $y=x+1$

3. $y=x-\left(2+3\sqrt{2}\right)$

4. $y=x-8$

Q. The distance between the lines $12x-5y+20=0$ and $12x-5y-6=0$ is

1. $5\text{units}$
2. $4\text{units}$
3. $2\text{units}$
4. $1\text{unit}$

Q. If a normal to the hyperbola $xy=c^2$ at $(c{t}_1,\frac{c}{t_1})$ meets the curve again at $(c{t}_2,\frac{c}{t_2})$ then:

1. 
2. 
3. 
4. 

Q.

For the equation $l{x}^2+mx+n=0,l\ne 0$. If $\alpha \&\beta$ are roots of the equation $m^3+l^{2}n+{\ln }^2=3lmn$ then

1. $\alpha ={\beta }^2$

2. ${\alpha }^3=\beta$

3. $\alpha +\beta =\alpha \beta$

4. $\alpha \beta =1$

Q.

The equation of two sides of a square are $5x-12y-65=0$ and $5x-12y+26=0$. Find the area of the square.

Q. A particle is moving in a straight line and at some moment it occupied the positions $(5,2)$ and $(-1,-2)$. Then the position of the particle when it is on $x-$axis is

1. $(-2,0)$
2. $(0,2)$
3. $(2,0)$
4. $(4,0)$

Q.

Determine the distance between the following pair of parallel lines:

(i) $4x-3y-9=0$ and $4x-3y-24=0$

(ii) $8x+15y-34=0$ and $8x+15y+31=0$

(iii) $y=mx+c$ and $y=mx+d$

(iv) $4x+3y-11=0$ and $8x+6y=15$

Q. The area of a square whose two sides lie on lines x+y=1 and x+y+2=0 will be .

1. $\frac{9}{2}sq.units$
2. $\frac{9}{4}sq.units$
3. $\frac{3}{2}sq.units$
4. $\frac{3}{4}sq.units$

Q. Find the distance between the parallel lines $3x-4y+9=0$ and $6x-8y-15=0$.

Q. Consider the lines represented by the equation $(x^2+xy-x)(x-y)=0$, forming a triangle, then
$\begin{array}{cc}Column1& Column2\\ \left(a\right)Orthocentreoftraingle& (\frac{1}{6},\frac{1}{2})\\ \left(b\right)Circumcenter& \left(\frac{1}{2+2\sqrt{2}}\right)\\ \left(c\right)Centroid& (0,\frac{1}{2})\\ \left(d\right)Incenter& (\frac{1}{2},\frac{1}{2})\end{array}$

1. a-s, b-r, c-q, d-p1

2. a-s, b-r, c-p, d-q

3. b-r, b-s, c-q, d-p

4. a-r, b-s, c-p, d-q

Q. The equation of line midway between the lines $2x-y+3=0,y=2x-4$ and parallel to them is.

Q. A line through $(0,-2)$ and is perpendicular to the line $5x-y+7=0$. Then the equation of the line parallel to it and at a distance of $3$ units from it is/are

1. $x+5y+10-3\sqrt{26}=0$
2. $x+5y+10-\sqrt{13}=0$
3. $x+5y+10+3\sqrt{26}=0$
4. $x+5y+10+\sqrt{13}=0$

Q.

Check whether the pair of lines $7x+y+3=0$ and $14x+2y+5=0$ is parallel or not.

Q. A line passing through the point $A(-6,5)$ meets the lines $x+3y+2=0$, $2x+y+4=0$ and $x-y-5=0$ at $P,Q$ and $R$ respectively. If ${\left(\frac{11}{AP}\right)}^2+{\left(\frac{16}{AR}\right)}^2={\left(\frac{3}{AQ}\right)}^2$,
then slope of the line can be,

1. $\pm \frac{\sqrt{2}}{3}$
2. $\pm 1$
3. $\pm \frac{\sqrt{3}}{2}$
4. $\pm \frac{2}{5}$