Equation of a Plane Parallel to a Given Plane

Q.

Equation of Plane which is parallel to XY plane is -

1. x = a ; where a is constant.

2. y = b ; where b is constant.

3. z = c ; where c is constant.

4. (x - a) + (y - b) = c ; where a, b, and c are constants.

Q.

Let the position vectors of points ‘$A$’ and ‘$B$’ be $\hat{i}+\hat{j}+\hat{k}$ and $2\hat{i}+\hat{j}+3\hat{k}$ respectively. A point ‘$P$’ divides the line segment $AB$ internally in the ratio $\lambda :1$ $\left(\lambda >0\right)$. If $O$ is the origin and $\overrightarrow{OB}·\overrightarrow{OP}-3{\left|\overrightarrow{OA}\times \overrightarrow{OP}\right|}^2=6$, then $\lambda$ is equal to:

Q.

The equation of the plane which bisects the angle between the planes 3x - 6y + 2z + 5 = 0 and 4x - 12y + 3z = 0 which contains the origin is

1. 
   

2. 
   

3. None of these

4. 
   

Q. A plane $P$ meets the coordinate axes at $A$, $B$ and $C$ respectively. The centroid of $△ABC$ is given to be $(1,1,2)$. Then the eqaution of the line through this centroid and perpendicular to the plane $P$ is

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

Q.

Foot of the perpendicular drawn from the origin to the plane $2-3y+4z=29$ is

1. $(5,-1,4)$

2. $(7,-1,3)$

3. $(5,-2,3)$

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

5. $(1,-3,4)$

Q.

The order of differential equations of all parabolas having directrix parallel to the $x$-axis is

1. $3$

2. $1$

3. $4$

4. $2$

Q. The equation of the plane which is parallel to the plane x-2y+2z=5 and whose distance from the point (1, 2, 3) is 1, is

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

Q. Find the equation of the plane passing through ( a , b , c ) and parallel to the plane

Q. The plane which is parallel to plane 3x - 3y +4z =7 is .

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

Q.

The equation $x^2+y^2+z^2=0$ represents

1. A circle

2. A plane

3. None of these

4. (0, 0, 0)

Q. There are ‘p’ points in space of which ‘q’ points are coplanar. Then the number of planes formed is

1. 
2. 
3. 
4. 

Q. The planes $2x-y+4z=5$ and $5x-2.5y+10z=6$ are

1. Perpendicular
2. Intersect y-axis
3. Parallel
4. Passes through $(0,0,\frac{5}{4})$
   

Q. The equation of the plane which makes with the coordinate axes, a triangle with centroid $(\alpha ,\beta ,\gamma )$ is given by

1. $\alpha x+\beta y+\gamma z=1$
2. $\alpha x+\beta y+\gamma z=3$
3. $\frac{x}{\alpha }+\frac{y}{\beta }+\frac{z}{\gamma }=1$
4. $\frac{x}{\alpha }+\frac{y}{\beta }+\frac{z}{\gamma }=3$

Q. A point $P$ moves on a plane $\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1$ and $O$ be the origin. A plane through $P$ and perpendicular to $OP$ meets the coordinate axes at $A,B$ and $C$ respectively. Let $G$ be the centroid of the triangle $ABC$. If the planes through $A,B$ and $C$ parallel to the planes $x=0,y=0$ and $z=0$, respectively, intersect at $Q$, then

1. If $OA=OB=OC$, then $OG=\frac{1}{3}OP$
2. Locus of $Q$ is
   $\frac{1}{ax}+\frac{1}{by}+\frac{1}{cz}=\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}$
3. Locus of $Q$ is
   $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}$
4. If $OA=OB=OC$, then $OG=OP$

Q.

A homogeneous differential equation of the form can be solved by making the substitution

A. y = vx

B. v = yx

C. x = vy

D. x = v

Q.

The equation of the plane which is parallel to xy-plane and cuts intercept of length 3 from the z-axis is

1. 

2. 

3. 

4. 

Q. If a plane meets the coordinate axes in $A,B$ and $C,$ such that the centroid of $\Delta ABC$ is $(1,2,4)$ then the perpendicular distance from origin to the plane is

1. 12
2. $\frac{12}{\sqrt{21}}$
3. 7
4. $\frac{7}{\sqrt{21}}$

Q. If the normal of the plane makes angles $\frac{\pi }{4},\frac{\pi }{4}\text{and}\frac{\pi }{2}$ with positive $X$-axis and $Y$-axis and $Z$-axis respectively and the length of the perpendicular line segment from origin to the plane is $\sqrt{2}$, then the equation of the plane is:

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

Q.

Equation of plane which is parallel to XY-plane is

1. x = a ; where a is constant.

2. y = b ; where b is constant.

3. z = c ; where c is constant.

4. (x - a) + (y - b) = c ; where a, b, and c are constants.

Q. x + y − 6z = 0
x − y + 2z = 0
−3x + y + 2z = 0

Q. If the lines $\frac{x-1}{-3}=\frac{y-2}{-2k}=\frac{z-3}{2}\mathrm{and}\frac{x-1}{k}=\frac{y-2}{1}=\frac{z-3}{5}$ are perpendicular, find the value of k and, hence, find the equation of the plane containing these lines.

Q.

Sir, while doing problems with section formula..there was a problem in which i have to find the ratio of vector..which is dividing a line..AND MANY PUBLISHERS ARE TAKING THE RATIO K:1 HOW???

Q. The equations of two given planes are
$a_{1}x+b_{1}y+c_{1}z=d_1{a}_{2}x+b_{2}y+c_{2}z=d_2$
If both the planes are parallel then which of the following is true:

1. $\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}=\frac{d_1}{d_2}$
2. $\frac{a_2}{a_1}=\frac{b_2}{b_1}=\frac{c_2}{c_1}=\frac{d_1}{d_2}$
3. None of the above
4. $\frac{a_2}{a_1}=\frac{b_2}{b_1}=\frac{c_2}{c_1}\ne \frac{d_2}{d_1}$

Q. Find the equations of the line passing through the point (−1, 2, 1) and parallel to the line $\frac{2x-1}{4}=\frac{3y+5}{2}=\frac{2-z}{3}.$

Q. Find the differential equation of all the parabolas with latus rectum '4a' and whose axes are parallel to x-axis.

Q.

Equation of plane which is parallel to X - axis will be -

1. mx + nz = C

2. my + nz = C

3. mx + ny = C

4. None of these.

Q. 2x − y + 2z = 0
5x + 3y − z = 0
x + 5y − 5z = 0

Q. The equation of the plane which bisects the angle between the planes $3x-6y+2z+5=0$ and $4x-12y+3z-3=0$ which contains the origin is ?

1. $x-3y+z-5=0$
2. $33x+13y+32z+45=0$
3. $Noneofthese$
4. $33x-13y+32z+45=0$

Q. x + y + z = 0
x − y − 5z = 0
x + 2y + 4z = 0

Q. The non-parametric vector equation of a plane passing through a point whose position vector is $\overrightarrow{a}$ is parallel to $\overrightarrow{u}$ and $\overrightarrow{v}$ is

1. $[\overrightarrow{r},\overrightarrow{u},\overrightarrow{v}]=0$
2. $[\overrightarrow{r},\overrightarrow{a},\overrightarrow{u}\times \overrightarrow{v}]$
3. $[\overrightarrow{a},\overrightarrow{u},\overrightarrow{v}]=0$
4. $[\overrightarrow{r}-\overrightarrow{a},\overrightarrow{u},\overrightarrow{v}]=0$