Family of Planes Passing through the Intersection of Two Planes

Q. The equation of the plane through the intersection of the planes x+2y+3z-4=0, 4x+3y+2z+1=0 and passing through the origin will be

1. x+y+z=0
2. 17x+14y+11z=0
3. 7x+4y+z=0
4. 17x+14y+z=0

Q.

$a{x}^2+2hxy+b{y}^2=0$ always represents a pair of straight lines passing through the origin. If

Column 1 Column 2

a. $h^2$>ab 1. Lines are coincident

b. $h^2$=ab 2. Lines are real and distinct

c. $h^2$<ab 3. Lines are imaginary with real point of intersection i.e. (0, 0)

1. a-1 b-2 c-3

2. a-2 b-1 c-3

3. a-3 b-2 c-1

4. a-3 b-1 c-2

Q.

The equation of a plane passing through the line of intersection of the planes $x+2y+3z=2$ and $x-y+z=3$ and at a distance$\frac{2}{\sqrt{3}}$from the point (3, 1, -1) is

1. $5x-11y+z=17$

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

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

4. $x-\sqrt{2}y=1-2\sqrt{2}$

Q. The plane 4x + 7y + 4z + 81 = 0 is rotated through a right angle about its line of intersection with the plane
5x + 3y + 10z = 25. The equation of the plane in its new position is x – 4y + 6z = k, where k, is

1. 106
2. -89
3. 73
4. 37

Q.

The planes x-cy-bz=0, cx-y+az=0 and bx+ay-z=0 pass through a straight line, where a, b, c are non-zero constants. Then the value of $a^2+b^2+c^2+2abc$ is

1. -1

2. 2

3. 1

4. 0