Family of Planes Passing through the Intersection of Two Planes

Q.

Draw the graph of the following linear equations in two variables: $x-y=2$

Q. The vector equation of the plane passing through the intersection of the planes $\overrightarrow{r}\cdot (\hat{i}+\hat{j}+\hat{k})=1$ and $\overrightarrow{r}\cdot (\hat{i}-2\hat{j})=-2,$ and the point $(1,0,2)$ is

1. $\overrightarrow{r}\cdot (\hat{i}+7\hat{j}+3\hat{k})=\frac{7}{3}$
2. $\overrightarrow{r}\cdot (\hat{i}-7\hat{j}+3\hat{k})=\frac{7}{3}$
3. $\overrightarrow{r}\cdot (\hat{i}+7\hat{j}+3\hat{k})=7$
4. $\overrightarrow{r}\cdot (3\hat{i}+7\hat{j}+3\hat{k})=7$

Q.

If the equation of a plane $P$, passing through the intersection of the planes, $x+4y-z+7=0$ and $3x+y+5z=8$ is $ax+by+6z=15$ for some $a,b\in R$, then the distance of the point$(3,2,-1)$ from the plane $P$ is ________ units.

Q. The equation of the plane containing the line 2x - 5y + z = 3, x + y + 4z = 5 and parallel to the plane x + 3y + 6z = 1, is

1. 2x + 6y + 12z = 13
2. x + 3y + 6z = -7
3. x + 3y + 6z = 7
4. 2x + 6y + 12z = -13

Q.

The distance between the line vector $\overrightarrow{r}=2\hat{i}-2\hat{j}+3\hat{k}+\lambda (\hat{i}-\hat{j}+4\hat{k})$ and the plane vector $\overrightarrow{r·}\left(\hat{i}+5\hat{j}+\hat{k}\right)=5$ is

1. $\frac{10}{3\sqrt{3}}$

2. $\frac{10}{9}$

3. $\frac{10}{3}$

4. $\frac{3}{10}$

Q.

The equation of line passing through the point of intersection of lines $3x-2y-1=0$ and $x-4y+3=0$ and the point $\left(\mathrm{\pi },0\right)$ is:

1. $x-y=\mathrm{\pi }$

2. $x-y=\mathrm{\pi }(\mathrm{y}+1)$

3. $x-y=\mathrm{\pi }(1-y)$

4. $x+y=\mathrm{\pi }(1-y)$

Q.

The value of $\lambda$ if $\lambda x^2-10xy+12y^2+5x-16y-3=0$ represents a pair of straight lines is

1. $1$

2. $-1$

3. $2$

4. $-2$

Q.

The lines $15x-18y+1=0,12x+10y-3=0$and $6x+66y-11=0$ are

1. Parallel

2. Perpendicular

3. Concurrent

4. None of these

Q. The equation of a plane containing the line of intersection of the planes $2x-y-4=0$ and $y+2z-4=0$ and passing through the point $(1,1,0)$ is:

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

Q. Let $a$ and $b$ respectively be the points of local maximum and local minimum of the function $f\left(x\right)=2x^3-3x^2-12x.$
If $A$ is the total area of the region bounded by $y=f\left(x\right),$ the $x-$axis and the lines $x=a$ and $x=b,$ then $4A$ is equal to .

Q. The equation of the plane passing through the intersection of the planes 2x – 5y + z = 3 and x + y + 4z = 5 and
parallel to the plane x + 3y + 6z = 1 is x + 3y + 6z = k, where k is

1. 5
2. 3
3. 7
4. 2

Q. The angle between the planes represented by $2x^2-6y^2-12z^2+18yz+2zx+xy=0$ is

1. ${\cos }^{-1}\frac{16}{21}$
2. ${\cos }^{-1}\frac{17}{21}$
3. ${\cos }^{-1}\frac{19}{21}$
4. $\frac{\pi }{2}$

Q.

Find the vector and the Cartesian equation of the line that passes through the points (3, -2, -5), (3, - 2, 6).

Q. The vector equation of the plane through the line of intersection of the planes $x+y+z=1$ and $2x+3y+4z=5$ which is perpendicular to the plane $x-y+z=0$ is :

1. $\overrightarrow{r}\times (\hat{i}-\hat{k})+2=0$
2. $\overrightarrow{r}\times (\hat{i}+\hat{k})+2=0$
3. $\overrightarrow{r}\cdot (\hat{i}-\hat{k})-2=0$
4. $\overrightarrow{r}\cdot (\hat{i}-\hat{k})+2=0$

Q. If the equation of a plane $P$, passing through the intersection of the planes, $x+4y-z+7=0$ and $3x+y+5z=8$ is $ax+by+6z=15$ for some $a,b\in R,$ then the distance of the point $(3,2,-1)$ from the plane $P$ is

Q. A plane $P:ax+by+cz=1$ passes through the intersection of planes $\overrightarrow{r}\cdot (\hat{i}+\hat{j}+\hat{k})=-3$ and $\overrightarrow{r}\cdot (\hat{i}-\hat{j}+\hat{k})=2.$ If plane $P$ divides the line segment joining $M(3,0,2)$ and $N(0,3,-1)$ in the ratio $2:1$ internally, then $(a+b+c)$ is equal to

Q. Find the equation of the plane which contains the line of intersection of the planes.
$\overrightarrow{r}.(\hat{i}-2\hat{j}+3\hat{k})-4=0$ and
$\overrightarrow{r}(-2\hat{i}+\hat{j}+\hat{k})+5=0$
and whose intercept on x-axis is equal to that of on y-axis.

Q. Let $p_1:x-2y+3z=5$ and $p_2:2x+3y+z+4=0$ be two planes. If $P$ is the foot of the perpendicular dropped from the origin $O$ to the line of intersection of the planes, then

1. $\overrightarrow{OP}\cdot ((\hat{i}-2\hat{j}+3\hat{k})\times (2\hat{i}+3\hat{j}+\hat{k}))=0$
2. if $Q$ is another point on the line of intersection of planes, then equation of the plane containing a triangle $OPQ$ is $14x+7y+17z=0.$
3. equation of the plane perpendicular to the line of intersection of the planes and passing through $(1,1,1)$ is $11x-5y-7z+1=0.$
4. If $N_1$ and $N_2$ are foot of the perpendicular from the origin $O$ to the planes $p_1$ and $p_2$ respectively, then $O{N}_1+O{N}_2$ is equal to $\frac{9}{\sqrt{14}}.$

Q. The plane of intersection of $x^2+y^2+z^2+2x+2y+2z+2=0$ and $4x^2+4y^2+4z^2+4x+4y+4z-1=0$ is:

1. $x+y+z+9=0$
2. $3x+3y+3z+4=0$
3. $4x+4y+4z+9=0$
4. They don't intesect

Q. If the distance between parallel planes $2x-y+3z-4=0$ and $6x-3y+9z+13=0$ is $D$. Then the value of $126D^2=$

Q. A straight line joining the points $(1,1,1)$ and $(0,0,0)$ intersects the plane $2x+2y+z=10$ at

1. $(1,2,5)$
2. $(2,2,2)$
3. $(2,1,5)$
4. $(1,1,6)$

Q.

If $y=4x+3$ is parallel to the tangent of the parabola $y^2=12x$, then its distance from the normal parallel to the given line is?

1. $\frac{213}{\sqrt{17}}$

2. $\frac{219}{\sqrt{17}}$

3. $\frac{211}{\sqrt{17}}$

4. $\frac{210}{\sqrt{17}}$

Q. The distance of the point $(2,1,-1)$ from the plane $x-2y+4z=9$ is $\frac{m}{\sqrt{21}}$ then $m$ is

Q. Write the sum of intercepts cut off by the plane $\overrightarrow{r}.(2\hat{i}+\hat{j}-\hat{k})-5=0$ on the three axes.

Q. A variable plane which remains at a constant distance $3p$ from the origin cuts the coordinate axes at $A,B,C$. Show that the locus of the centroid of triangle $ABC$ is $\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}=\frac{1}{p^2}$

Q. The vector equation of the line passing through the point $(-1,-1,2)$ and parallel to the line $2x-2=3y+1=6z-2,$ is

1. $\overrightarrow{r}=(-\hat{i}-\hat{j}+2\hat{k})+\lambda (3\hat{i}+2\hat{j}+\hat{k})$
2. $\overrightarrow{r}=(\hat{i}+\hat{j}-2\hat{k})+\lambda (3\hat{i}+2\hat{j}+\hat{k})$
3. $\overrightarrow{r}=(-\hat{i}-\hat{j}+2\hat{k})+\lambda (\hat{i}+2\hat{j}+3\hat{k})$
4. $\overrightarrow{r}=(-\hat{i}-\hat{j}+2\hat{k})+\lambda (3\hat{i}+2\hat{j}-\hat{k})$

Q. Consider a region $R=\left\{\right(x,y)\in R^2:x^2\le y\le 2x\}$. If a line $y=\alpha$ divides the area of region $R$ into two equal parts, then which of the following is true

1. ${\alpha }^3-6{\alpha }^2+16=0$
2. $3{\alpha }^2-8{\alpha }^{\frac{3}{2}}+8=0$
3. ${\alpha }^3-6{\alpha }^{\frac{3}{2}}-16$
4. $3{\alpha }^2-8\alpha +8=0$

Q.

In $R^3$, consider the planes $P_1:y=0$ and $P_2:x+z=1$. Let $P_3$ be a plane , different from $P_1$ and $P_2$, which passes through the intersection of $P_1$ and $P_2$. If the distance of the point (0, 1, 0) from
$P_3$ is 2, then which of the following relation(s) is/are true?

1. $2\alpha +\beta +2\gamma +2=0$

2. $2\alpha -\beta +2\gamma +4=0$

3. $2\alpha +\beta -2\gamma -10=0$

4. $2\alpha -\beta +2\gamma -8=0$

Q. The point $(3,-5,6)$ lies in

1. $II$ Octant
2. $III$ Octant
3. $IV$ Octant
4. $VI$ Octant

Q. The equation of the plane which contains the line of intersection of planes $x+2y+3z-4=0$ and $2x+y-z+5=0$ which is perpandicular to the plane $5x+3y-6z+8=0$

1. $3x+4y+5z-14=0$
2. $33x+45y+50z-41=0$
3. $37x+51y+49z-45=0$
4. $7x+8y+15z-19=0$