If the equation of the plane passing through the line of intersection of the planes $2x - y + z = 3$ and $4x - 3y + 5z + 9 = 0$ and parallel to the line $\frac{x + 1}{-2} = \frac{y + 3}{4} = \frac{z - 2}{5}$ is $ax + by + cz + 6 = 0$,then $a + b + c$ is equal to $.............$.

The position vector of the point in which the line joining the points $i - 2j + k$ and $3k - 2j$ cuts the plane through the origin and the points $4j$ and $2i + k$,is

The plane $2x - y + 3z + 5 = 0$ is rotated through $90^o$ about its line of intersection with the plane $5x - 4y - 2z + 1 = 0$. The equation of the plane in its new position is:

$l, m, n$ are three unit vectors in a right-handed system and $L$ is a line through the points $A, B, C$ whose position vectors are $p l + 7 m - 6 n, 2 l + 5 m - 4 n$ and $l + 4 m - 3 n$ respectively. If the equation of the plane containing $L$ and the point $(-p, p, p+1)$ is $ax + by + cz = 1$,then $p(a+b+c) =$

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

If the line $\frac{x-3}{2}=\frac{y+2}{-1}=\frac{z+4}{3}$ lies in the plane $\ell x+m y-z=9$,then $\ell^2+m^2$ is