Condition for Two Lines to be on the Same Plane

Q.

If the slope of one of the lines represented by $a{x}^2+2hxy+b{y}^2=0$ is the square of the other, then $\frac{a+b}{h}+\frac{8h^2}{ab}$is

1. $3$

2. $4$

3. $5$

4. $6$

Q. $\overrightarrow{a}$ and $\overrightarrow{b}$ are non-zero non-collinear vectors such that |$\overrightarrow{a}$|=2, $\overrightarrow{a}.\overrightarrow{b}$=1 and angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is $\frac{\pi }{3}$. If $\overrightarrow{r}$ any vector satisfying $\overrightarrow{r}$.
$\overrightarrow{a}=2,\overrightarrow{r}.\overrightarrow{b}$ = 8, $(\overrightarrow{r}+2\overrightarrow{a}-10).(\overrightarrow{a}\times \overrightarrow{b})$ = 6 and is equal to $\overrightarrow{r}+2\overrightarrow{a}-10\overrightarrow{b}=|(\overrightarrow{a}\times \overrightarrow{b})|$= then I=

1. 
2. 
3. 3
4. 2

Q.

The equation of the line joining the origin to the point $(-4,5)$ is:

1. $5x+4y=0$

2. $3x+4y=2$

3. $5x-4y=0$

4. $4x-5y=0$

Q. Out of the given four alternatives, three are similar in a certain way. Choose the figure which is different from the rest.

1. 
   
2. 
   
3. 
   
4. 
   

Q.

If the straight lines $\frac{x-1}{2}=\frac{y+1}{K}=\frac{z}{2}$ and $\frac{x+1}{5}=\frac{y+1}{2}=\frac{z}{K}$ are coplanar, then the plane(s) containing these two lines is/are

1. y +2z = - 1

2. y + z = - 1

3. y - z = - 1

4. y - 2z = - 1

Q.

If the lines $x+y=a$ and $x-y=b$ touch the curve $y=x^2-3x+2$ at the points where the curve intersects the $x-$axis, then $\frac{a}{b}$ is equal to

Q.

If the lines $\frac{x-2}{1}=\frac{y-3}{1}=\frac{z-4}{-k}$ and$\frac{x-1}{k}=\frac{y-4}{2}=\frac{z-5}{1}$ are coplanar, then k can have

1. exactly three values

2. exactly one value

3. exactly two values

4. any value

Q. The scalar product of the vector $\overrightarrow{a}=\hat{i}+\hat{j}+\hat{k}$ with a unit vector along the sum of vectors$\overrightarrow{b}=2\hat{i}+4\hat{j}-5\hat{k}\text{and}\overrightarrow{c}=\lambda \hat{i}+2\hat{j}+3\hat{k}$ is equal to one. Find the value of $\lambda$ and hence find the unit vector along $\overrightarrow{b}+\overrightarrow{c}$.

Q.

If the distance between the plane, $23x-10y-2z+48=0$ and the plane containing the lines $\frac{x+1}{2}=\frac{y-3}{4}=\frac{z+1}{3}$​ and $\frac{x+3}{2}=\frac{y+2}{6}=\frac{z-1}{\lambda }$$(\lambda \in \mathrm{ℝ})$ is equal to $\frac{k}{\sqrt{633}}$ then $k$ is equal to

Q.

The value of $\left[\overrightarrow{a}-\overrightarrow{b}\overrightarrow{b}-\overrightarrow{c}\overrightarrow{c}-\overrightarrow{a}\right]$ is ?

1. $0$

2. $1$

3. $2$

4. None of the above.

Q.

The equation $y^2-x^2+2x-1=0$ represents

1. A pair of straight lines

2. A circle

3. A parabola

4. An ellipse

Q.

If $a{x}^2-y^2+4x-y=0$ represents a pair of lines, then $a$ is equal to

1. $-16$

2. $16$

3. $-4$

4. $4$

Q.

Use the table to write a linear function that relates $\mathrm{y}$ to $\mathrm{x}$.

Q. The number of distinct real values of $\lambda$ for which the lines $\frac{x-1}{1}=\frac{y-2}{2}=\frac{z+3}{{\lambda }^2}$ and $\frac{x-3}{1}=\frac{y-2}{{\lambda }^2}=\frac{z-1}{2}$ are coplanar is:

1. $3$
2. $1$
3. $2$
4. $4$

Q. The lines $\frac{x-a+d}{\alpha -\delta }=\frac{y-a}{\alpha }=\frac{z-a-d}{\alpha +\delta }$ and $\frac{x-b+c}{\delta -\lambda }=\frac{y-b}{\beta }=\frac{z-b-c}{\beta +\gamma }$ are coplanar and then equation to the plane in which they lie, is

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

Q. If $\overrightarrow{x},\overrightarrow{y}$ are two non-zero and non-collinear vectors satisfying $\left[\right(a-2){\alpha }^2+(b-3)\alpha +c]\overrightarrow{x}+\left[\right(a-2){\beta }^2+(b-3)\beta +c]\overrightarrow{y}+\left[\right(a-2){\gamma }^2+(b-3)\gamma +c](\overrightarrow{x}\times \overrightarrow{y})=\overrightarrow{0},$ where $\alpha ,\beta ,\gamma$ are three distinct real numbers, then which of the following statement(s) is/are correct ?

1. $a(b+c-a)=2$
2. $b(c+a-b)=3$
3. $c(a+b-c)=0$
4. $a^2+b^2-c^2=13$

Q. If the vectors $\overrightarrow{\alpha }=\hat{i}+a\hat{j}+a^2\hat{k},\overrightarrow{\beta }=\hat{i}+b\hat{j}+b^2\hat{k}\text{and}\overrightarrow{\gamma }=\hat{i}+c\hat{j}+c^2\hat{k}$ are three non-coplanar vectors and $\left|\begin{array}{ccc}a& a^2& 1+a^3\\ b& b^2& 1+b^3\\ c& c^2& 1+c^3\end{array}\right|=0$ , then the value of $abc$ is

1. $2$
2. $1$
3. $0$
4. $-1$

Q. If $\overrightarrow{p}$ and $\overrightarrow{s}$ are not perpendicular to each other and $\overrightarrow{r}\times \overrightarrow{p}=\overrightarrow{q}\times \overrightarrow{p}$ and $\overrightarrow{r}\cdot \overrightarrow{s}=0,$ then $\overrightarrow{r}$ is equal to

1. $\overrightarrow{q}+\left(\frac{\overrightarrow{q}\cdot \overrightarrow{p}}{\overrightarrow{p}\cdot \overrightarrow{s}}\right)\overrightarrow{p}$
2. $\overrightarrow{q}+\mu \overrightarrow{p}$ for all $\mu$
3. $\overrightarrow{q}-\left(\frac{\overrightarrow{q}\cdot \overrightarrow{s}}{\overrightarrow{p}\cdot \overrightarrow{s}}\right)\overrightarrow{p}$
4. $\overrightarrow{p}\cdot \overrightarrow{s}$

Q.

If a and b are two arbitrary constants, then the straight line $(a-2b)x+(a+3b)y+3a+4b=0$ will pass through

1. $(–1,–2)$

2. $(1,2)$

3. $(–2,–3)$

4. $(2,3)$

Q.

The angle between the line joining the points $\left(1,-2\right),\left(3,2\right)$ and the line $x+2y-7=0$ is

1. $\mathrm{\pi }$

2. $\frac{\mathrm{\pi }}{2}$

3. $\frac{\mathrm{\pi }}{3}$

4. $\frac{\mathrm{\pi }}{6}$

Q. If $\overrightarrow{a},\overrightarrow{b}$ and $\overrightarrow{c}$ are unit vectors satisfying $|\overrightarrow{a}-\overrightarrow{b}{|}^2+|\overrightarrow{b}-\overrightarrow{c}{|}^2+|\overrightarrow{c}-\overrightarrow{a}{|}^2=9,$ then $|2\overrightarrow{a}+5\overrightarrow{b}+5\overrightarrow{c}|$ is

Q. If $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ are three unit vectors, then $|\overrightarrow{a}-\overrightarrow{b}{|}^2+|\overrightarrow{b}-\overrightarrow{c}{|}^2+|\overrightarrow{c}-\overrightarrow{a}{|}^2$ does not exceed

1. $4$
2. $9$
3. $8$
4. $6$

Q.

What does it mean for two planes to be parallel?

Q. If $\alpha \overrightarrow{a}+\beta \overrightarrow{b}+\gamma \overrightarrow{c}=\overrightarrow{0},$ then the resultant of $(\overrightarrow{a}\times \overrightarrow{b})\times \left\{\right(\overrightarrow{b}\times \overrightarrow{c})\times (\overrightarrow{c}\times \overrightarrow{a}\left)\right\}$ is
(where $\alpha ,\beta$ and $\gamma$ are not simultaneously zero.)

1. $\alpha -\beta +\gamma$
2. $(\alpha +\beta +\gamma )(\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c})$
3. $\gamma \overrightarrow{a}+\overrightarrow{b}$
4. $\overrightarrow{0}$

Q.

If the lines $\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-1}{4}$ and $\frac{x-3}{1}=\frac{y-k}{1}=\frac{z}{1}$ and intersect, then k =
[IIT Screening 2004]

1. $\frac{2}{9}$

2. $\frac{2}{9}$

3. $Noneofthese$

4. $0$

Q. A man is walking on a straight line. The arithmetic mean of the reciprocals of the intercepts of this line on the coordinate axes is $\frac{1}{4}$. Three stones $A,B\text{and}C$ are placed at the points $(1,1),(2,2)\text{and}(4,4)$ respectively. Then which of these stones is/are on the path of the man?

1. $B$only
2. $A$only
3. the three
4. $C$only

Q. The line $\frac{x-2}{1}=\frac{y-3}{1}=\frac{z-4}{-k}$ and $\frac{x-1}{k}=\frac{y-4}{2}=\frac{z-5}{1}$ are coplanar. if

1. k = 0 or -1
2. k = 0 or 1
3. k = 0 or -3
4. k = 3 or -3

Q. The straight line $\frac{x-2}{1}=\frac{y-3}{1}=\frac{z-4}{-k}$ and $\frac{x-1}{k}=\frac{y-4}{2}=\frac{z-5}{1}$ will intersect provided if k=:

1. k = {3, –3}
2. k = {–1, 1}
3. k = {0, –1}
4. k = {0, –3}

Q. If the vectors $a\hat{i}+\hat{j}+\hat{k},\hat{i}+b\hat{j}+\hat{k}$ and $\hat{i}+\hat{j}+c\hat{k}(a,b,c\ne 1)$ are coplanar, then the value of $\frac{1}{1-a}+\frac{1}{1-b}+\frac{1}{1-c}$ is equal to

1. $0$
2. $a+b+c$
3. $1$
4. $abc$

Q. The lines $\frac{x-a+d}{\alpha -\delta }=\frac{y-a}{\alpha }=\frac{z-a-d}{\alpha +\delta }$ and $\frac{x-b+c}{\beta -\gamma }=\frac{y-b}{\beta }=\frac{z-b-c}{\beta +\gamma }$ are coplanar and then equation to the plane in which they lie, is

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