Condition for a Line to Lie on a Plane

Q.

The pair of straight lines passing through the point $\left(1,2\right)$ and perpendicular to the pair of straight lines $3x^2-8xy+5y^2=0$, is

1. $\left(5x+3y+11\right)\left(x+y+3\right)$

2. $\left(5x+3y-11\right)\left(x+y-3\right)$

3. $\left(5x+3y-11\right)\left(x+y+3\right)$

4. $\left(5x+3y+11\right)\left(x+y-3\right)$

Q. Let a plane $P$ pass through the point $(3,7,-7)$ and contain the line, $\frac{x-2}{-3}=\frac{y-3}{2}=\frac{z+2}{1}$. If distance of the plane $P$ from the origin is $d,$ then $d^2$ is equal to

Q.

The equation $2x^2+4xy-p{y}^2+4x+qy+1=0$ will represent two mutually perpendicular straight lines, if

1. $p=1andq=2or6$

2. $p=2andq=0or6$

3. $p=2andq=0or8$

4. $p=–2andq=–2or8$

Q.

A straight line passes through a fixed point $(h,k)$. The locus of the foot of perpendicular on it drawn from the origin is

1. $x^2+y^2–hx–ky=0$

2. $x^2+y^2+hx+ky=0$

3. $3x^2+3y^2+hx–ky=0$

4. None of these

Q. The plane containing the two lines $\frac{x-3}{1}=\frac{y-2}{4}=\frac{z-1}{5}and\frac{x-2}{1}=\frac{y+3}{-4}=\frac{z+1}{5}$ is 11x + my + nz = 28, where

1. m = –1, n = 3
2. m = 1, n = –3
3. m = –1, n = –3
4. m = 1, n = 3

Q. Let $A$ be a point on the line $\overrightarrow{r}=(1-3\mu )\hat{i}+(\mu -1)\hat{j}+(2+5\mu )\hat{k}$ and $B(3,2,6)$ be a point in the space. Then the value of $\mu$ for which the vector $\overrightarrow{AB}$ is parallel to the plane $x-4y+3z=1$ is :

1. $\frac{1}{4}$
2. $\frac{1}{8}$
3. $-\frac{1}{4}$
4. $\frac{1}{2}$

Q. Let the line $\frac{x-2}{3}=\frac{y-1}{-5}=\frac{z+2}{2}$ lies in the plane $x+3y-\alpha z+\beta =0.$ Then $(\alpha ,\beta )$ equals

1. (6, - 17)
2. (-6, 7)
3. (5, -15)
4. (-5, 15)

Q. A line $L$ passes through a point with position vector $\overrightarrow{a}$ and is perpendicular to the plane containing the two intersecting lines $\overrightarrow{r}=\overrightarrow{b}+\lambda \overrightarrow{p}$ and $\overrightarrow{r}=\overrightarrow{c}+\mu \overrightarrow{q},$ where $\lambda ,\mu \in R.$ If $L$ intersects a plane $\overrightarrow{r}\cdot \overrightarrow{n}=d$ at the point $R,$ then the position vector of the point $R$ is

1. $(d-\overrightarrow{a}\cdot \overrightarrow{n})(\overrightarrow{p}\times \overrightarrow{q})$
2. $\overrightarrow{a}+\frac{(d-\overrightarrow{a}\cdot \overrightarrow{n})}{\left[\overrightarrow{p}\overrightarrow{q}\overrightarrow{n}\right]}(\overrightarrow{p}\times \overrightarrow{q})$
3. $\overrightarrow{a}+\left[\overrightarrow{p}\overrightarrow{q}\overrightarrow{n}\right]\overrightarrow{b}+\left[\overrightarrow{p}\overrightarrow{q}\overrightarrow{a}\right]\overrightarrow{c}$
4. $\overrightarrow{a}+\frac{(d+\overrightarrow{a}\cdot \overrightarrow{n})}{\left[\overrightarrow{p}\overrightarrow{q}\overrightarrow{n}\right]}(\overrightarrow{p}\times \overrightarrow{q})$

Q.

If the line, $\frac{x-3}{2}=\frac{y+2}{-1}=\frac{z+4}{3}$ lies in the plane, lx+my-z = 9, then $l^2+m^2$ is equal to

1. 18

2. 26

3. 2

4. 5

Q. A unit vector normal to the plane through the points $\hat{i},2\hat{j}$ and $3\hat{k}$ is

1. $\frac{-6\hat{i}+3\hat{j}+2\hat{k}}{7}$
2. $\frac{6\hat{i}+3\hat{j}-2\hat{k}}{7}$
3. $\frac{6\hat{i}+3\hat{j}+2\hat{k}}{7}$
4. $\frac{6\hat{i}-3\hat{j}+2\hat{k}}{7}$

Q. The direction ratios of two lines $L_1$ and $L_2$ are $<4,-1,3>$ and $<2,-1,2>$ respectively. A vector $\overrightarrow{V}$ is perpendicular to $L_1$ and $L_2$ both such that $|\overrightarrow{V}|=15$. If $\overrightarrow{V}=x_1\hat{i}+x_2\hat{j}+x_3\hat{k},$ then the value of $|x_1+x_2+x_3|$ is

Q. If the line $\frac{x-2}{3}=\frac{y+1}{2}=\frac{z-1}{-1}$ intersects the plane $2x+3y-z+13=0$ at a point $P$ and the plane $3x+y+4z=16$ at a point $Q,$ then $PQ$ is equal to :

1. $2\sqrt{14}$
2. $14$
3. $2\sqrt{7}$
4. $\sqrt{14}$

Q. Line of intersection of the planes $x+2y=0$ and $y-3z+3=0$ is

1. $\frac{x}{-6}=\frac{y}{3}=\frac{z}{1}$
2. $\frac{x}{2}=\frac{y-3}{-1}=\frac{z}{1}$
3. $\frac{x+6}{-2}=\frac{y-3}{1}=\frac{-z-2}{1}$
4. $\frac{x+6}{-6}=\frac{y-3}{3}=\frac{z-2}{1}$

Q. A line is drawn from the point P(1, 1, 1) and perpendicular to a line with direction ratios (1, 1, 1) to intersect the plane x+2y+3z=4 at Q The locus of point Q is

1. $\frac{x}{-2}=\frac{y-5}{1}=\frac{z+2}{1}$
2. $\frac{x}{1}=\frac{y-5}{-2}=\frac{z+2}{1}$
3. $x=y=z$
4. $\frac{x}{2}=\frac{y}{3}=\frac{z}{5}$

Q. $Q\left(Z_q\right)$ is the foot of perpendicular drawn from $P\left(Z_p\right)$ to the line $a\overline{z}+z\overline{a}+b=0,bϵR$ and ‘a’ is complex number, then:

1. 
2. 
3. 
4. 

Q. The d.r's of two parallel lines are $4,-3,-1$ and $\lambda +\mu ,1+\mu ,2$ then $(\lambda ,\mu )$ is:

1. $(1,7)$
2. $(-1,-7)$
3. $(7,1)$
4. $(-13,5)$

Q. The shortest distance between the lines $\frac{x-1}{0}=\frac{y+1}{-1}=\frac{z}{1}$ and $x+y+z+1=0,2x-y+z+3=0$ is:

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

Q. The vector equation of the straight line passing through the point with position vector $\hat{i}-3\hat{j}+\hat{k}$ and parallel to the vector, $2\hat{i}+3\hat{j}-4\hat{k}$ in standard cartesian form is:

1. $\frac{x+1}{2}=\frac{y+3}{3}=\frac{z-1}{-4}$
2. $\frac{x-1}{2}=\frac{y+3}{3}=\frac{z-1}{4}$
3. $\frac{x-1}{2}=\frac{y+3}{3}=\frac{z-1}{-4}$
4. $\frac{x-1}{2}=\frac{y-3}{3}=\frac{z+1}{-4}$

Q. If $A,B,C$ and $D$ be four distinct points in space such that $\overrightarrow{AB}$ is not perpendicular to $\overrightarrow{CD}$ and satisfies $\overrightarrow{AB}\cdot \overrightarrow{CD}=\frac{1}{\lambda }\left(\right(\overrightarrow{AD}{)}^2+(\overrightarrow{BC}{)}^2-(\overrightarrow{AC}{)}^2-\left(\overrightarrow{BD}{)}^2\right),$ then the value of $\lambda$ is

Q.

A line and parabola lies on the same plane.

(1) Line may never touch the parabola

(2) Line may touch parabola at a single point

(3)Line may touch the parabola at one point and intersect at a different point

(4)Line may cut the parabola at 2 or more points

1. Line may never touch the parabola

2. Line may cut the parabola at 2 or more points

3. Line may touch parabola at a single point

4. Line may touch the parabola at one point and intersect at a different point

Q. If the distance of the point $P(1,–2,1)$ from the plane $x+2y–2z=\alpha$, where $\alpha >0$, is $5$, then the foot of the perpendicular from $P$ to the plane is

1. $(\frac{2}{3},-\frac{1}{3},\frac{5}{3})$
2. $(\frac{1}{3},\frac{2}{3},\frac{10}{3})$
3. $(\frac{4}{3},-\frac{4}{3},\frac{1}{3})$
4. $(\frac{8}{3},\frac{4}{3},-\frac{7}{3})$

Q. If $\overrightarrow{x}$ is a vector whose initial point divides the line joining $5\hat{i}$ and $5\hat{j}$ in the ratio $\lambda :1$ and the terminal point is the origin. Also $|\overrightarrow{x}|\le \sqrt{37},$ then $\lambda$ belongs to

1. $(-\infty ,-\frac{1}{6})$
2. $(-\infty ,-6]\cup [-\frac{1}{6},\infty )$
3. $(-\frac{1}{6},\frac{1}{6})$
4. $(-6,-\frac{1}{6})$

Q. From a point $P(\lambda ,\lambda ,\lambda )$, perpendiculars $PQ$ and $PR$ are drawn respectively on the lines $y=x,z=1$ and $y=-x,z=-1$. If $P$ is such that $\angle QPR$ is a right angle, then the possible value(s) of $\lambda$ is(are)

1. $\sqrt{2}$
2. $1$
3. $-1$
4. $-\sqrt{2}$

Q. Find the coordinate of the point $P$ where the line through $A(3,-4,-5)$ and $B(2,-3,1)$ crosses the plane passing through three points $L(2,2,1),M(3,0,1)$ and $N(4,-1,0)$. Also, find the ratio in which $P$ divides the line segment $AB$.

Q. Let P be the image of the point $(3,1,7)$ with respect to the plane $x-y+z=3$. Then the equation of the plane passing through P and containing the straight line
$\frac{x}{1}=\frac{y}{2}=\frac{z}{1}$ is

1. $x+y-3z=0$
2. $3x+z=0$
3. $x-4y+7z=0$
4. $2x-y=0$

Q.

Let the line
$\frac{x-2}{3}=\frac{y-1}{-5}=\frac{z+2}{2}$
lies in the plane $x+3y-\alpha z+\beta =0.$ Then $(\alpha ,\beta )$ equals

1. $(-6,7)$

2. $(5,-15)$

3. $(-5,15)$

4. $(6,-17)$

Q. Let $R$ be the real line. Consider the following subsets of the plane $R\times R$
$S=\{x,y\}:y=x+1$ and $0<x<2$
$T=\{x,y\}:x-y$ is an integer
Which one of the following is true?

1. Neither $S$ nor $T$ is an equivalence relation on $R$
2. Both $S$ and $T$ are equivalence relations on $R$
3. $S$ is an equivalence relation on $R$ but $T$ is not
4. $T$ is an equivalence relation on $R$ but $S$ is not

Q.

Let the line
$\frac{x-2}{3}=\frac{y-1}{-5}=\frac{z+2}{2}$
lies in the plane $x+3y-\alpha z+\beta =0.$ Then $(\alpha ,\beta )$ equals

1. $(6,-17)$

2. $(-6,7)$

3. $(5,-15)$

4. $(-5,15)$

Q. $\mathrm{Line}\frac{x-1}{5}=\frac{y+5}{-2}=\frac{z+1}{-1}\mathrm{lies}\mathrm{on}\mathrm{the}\mathrm{plane}2x+5y-z+5=0$

1. True
2. False

Q. For the given lines $\overrightarrow{r}=-\hat{i}+4\hat{j}+2\hat{k}+\lambda (\hat{i}-2\hat{j}+\hat{k})$ and $\overrightarrow{r}=2\hat{i}+3\hat{j}+\hat{k}+\mu (\hat{i}+3\hat{j}-3\hat{k}),$ which of the following is correct ?

1. Lines are parallel
2. Lines are coincident
3. Lines are coplanar
4. Lines are skew