Equation of a Plane : General Form

Q. Let the equation of the pair of lines, $y=px$ and $y=qx$, can be written as $(y-px)(y-qx)=0$. Then the equation of the pair of the angle bisectors of the lines $x^2-4xy-5y^2=0$ is

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

Q. A variable plane which remains at a constant distance p from the origin cuts the coordinate axes in A, B, C. The locus of the centroid of the tetrahedron OABC is $y^2{z}^2+z^2{x}^2+x^2{y}^2=k{x}^2{y}^2{z}^2$, where k is equal to

1. $9p^2$
2. $\frac{9}{p^2}$
3. $\frac{16}{p^2}$
4. $\frac{7}{p^2}$

Q. A piece of cheese is located at (12, 10) in a coordinate plane. A mouse is at (4, –2) and is running up the line y = –5x + 18. At the point (a, b), the mouse starts getting farther from the cheese rather than closer to it. The value of (a + b) is

1. 6
2. 10
3. 18
4. 14

Q. If a plane passes through the point (1, 1, 1) and is perpendicular to the line $\frac{x-1}{3}=\frac{y-1}{0}=\frac{z-1}{4}$ then its perpendicular distance from the origin is

1. $1$
2. $\frac{3}{4}$
3. $\frac{4}{3}$
4. $\frac{7}{5}$

Q. If the general equation of plane is given by ax + by + cz = d then a, b, c are the of the to the plane.

1. normal
2. tangent
3. direction cosines
4. direction ratios

Q. Consider the three planes
$P_1:3x+15y+21z=9$
$P_2:x-3y-z=5,$ and
$P_3:2x+10y+14z=5$
Then, which one of the following is true ?

1. $P_1$ and $P_3$ are parallel.
2. $P_2$ and $P_3$ are parallel.
3. $P_1,P_2$ and $P_3$ all are parallel.
4. $P_1$ and $P_2$ are parallel.

Q.

The equation of the plane passing through the points (1, -1, 2) and (2, -2 2) and which is perpendicular to the plane 6x -2y +2z =9 is

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

2. $x-y-2z=4$

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

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

Q.

Draw the graph of the following linear equations in two variables: $x+y=4$

Q. An equation of the plane passing through the point (1, -1, 2)and parallel to the plane 3x + 4y - 5z = 0 is

1. 3x+4y-5z=11
2. 3x+4x-5z+11=0
3. 6x+8y-10z=1
4. 3x+4y-5z=2

Q.

The equation of the plane passing through the lines $\frac{x-4}{1}=\frac{y-3}{1}=\frac{z-2}{2}$ and $\frac{x-3}{1}=\frac{y-2}{-4}=\frac{z}{5}$ is

1. $11x+y-3z=35$

2. $11x-y-3z=35$

3. $11x-y+3z=35$

4. None of these

Q.

A point is moves on xy plane.If the sum of the distance from two mutual perpendicular lines is 5 then area under it is

Q. The least positive value of $t,$ so that the lines $x=t+\alpha ,y+16=0$ and $y=\alpha x$ are concurrent, is:

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

Q. If O be the origin and the coordinates of P be (1, 2, −3), then find the equation of the plane passing through P and perpendicular to OP.

Q. Reflection of the line $\frac{x-1}{-1}=\frac{y-2}{3}=\frac{z-4}{1}$ in the plane x +y +z =7 is :
​

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

Q. The equation of the plane containing the two lines of intersection of the two pairs of planes x + 2y – z – 3 = 0 and 3x – y + 2z – 1 = 0, 2x – 2y + 3z = 0 and x – y + z + 1 =0 is :

1. None of these
2. 7x – 7y + 8z + 3 = 0
3. 7x – 8y + 9z + 3 = 0
4. 5x – 5y + 6z + 2 = 0

Q.

Find the equations of the planes that passes through three points.

(a) (1, 1, −1), (6, 4, −5), (−4, −2, 3)

(b) (1, 1, 0), (1, 2, 1), (−2, 2, −1)

Q. Find the equation of the plane that contains the point $(1,–1,2)$ and is perpendicular to each of the planes $2x+3y–2z=5$ and $x+2y–3z=8$.

Q.

Find the equation of the parabola that satisfies the following conditions: Vertex (0, 0) passing through (2, 3) and axis is along x-axis

Q. Reflection of the line $\frac{x-1}{-1}=\frac{y-2}{3}=\frac{z-4}{1}$ in the plane x +y +z =7 is :
​

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

Q. 30. P is a point(a, b, c) . Let A , B , C be images of P in y_z , z_x and x_y planes respectively , then the equation of the plane ABC is

Q. If lx+my+nz = d is the general form of the equation of a plane and l, m, n are direction cosines of that plane, then what does d represent?

1. The normal distance of the plane from the origin
2. Direction cosine of the z-axis
3. Direction cosine of the y-axis
4. Direction cosine of the x-axis

Q. Find the Cartesian equation of the following planes: (a) (b) (c)

Q.

If l, m, n are the direction cosines of the normal to the plane and p be the perpendicular distance of the plane from the origin, then the equation of the plane is:

1. 

2. 

3. 

4. 

Q.

Find the area of the region bounded by the ellipse

Q. The distance of the point (1, –2, 3) from the plane x – y + z = 5 measured parallel to the line $\left(\frac{1}{2}\right)x=\left(\frac{1}{3}\right)y=\left(\frac{-1}{6}\right)z$ is :

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

Q. Number of solutions for the system of equations $x+2y+z=0,3x+5y+2z=2$ and $2x+4y+2z=1$

1. $1$
2. infinity
3. $2$
4. $0$

Q.

What is the cartesian equation of a plane?

Q.

If the general equation of plane is given by ax + by + cz = d then a, b, c are the direction ratios of the normal to the plane.

1. True

2. False

Q.

What is stationary point in maxima and minima?

Q. Find the area of the region bounded by y = $\sqrt{x}$ and y = x.