The direction ratios of the normal to the pla | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(D) The equation of the family of planes passing through the line of intersection of the planes $x+2y+3z-4=0$ and $4x+3y+2z-1=0$ is given by $(x+2y+3z

Vedclass · Accepted Answer

$3, 2, 1$

Vedclass · Answer

(D) The equation of the family of planes passing through the line of intersection of the planes $x+2y+3z-4=0$ and $4x+3y+2z-1=0$ is given by $(x+2y+3z-4) + \lambda(4x+3y+2z-1) = 0$. Since the plane passes through the origin $(0, 0, 0)$,we substitute $x=0, y=0, z=0$ into the equation: $(0+0+0-4) + \lambda(0+0+0-1) = 0 \Rightarrow -4 - \lambda = 0 \Rightarrow \lambda = -4$. Substituting $\lambda = -4$ back into the family equation: $(x+2y+3z-4) - 4(4x+3y+2z-1) = 0 \Rightarrow x+2y+3z-4 - 16x - 12y - 8z + 4 = 0 \Rightarrow -15x - 10y - 5z = 0 \Rightarrow 3x + 2y + z = 0$. The direction ratios of the normal to the plane $ax+by+cz+d=0$ are $(a, b, c)$. Thus,the direction ratios of the normal to the plane $3x+2y+z=0$ are $3, 2, 1$.

The direction ratios of the normal to the plane passing through the origin and the line of intersection of the planes $x+2y+3z=4$ and $4x+3y+2z=1$ are . . . . . .

Similar Questions

The distance between two parallel planes $2x + y + 2z = 8$ and $4x + 2y + 4z + 5 = 0$ is:

In the following cases,determine whether the given planes are parallel or perpendicular,and in case they are neither,find the angles between them.
$2x + y + 3z - 2 = 0$ and $x - 2y + 5 = 0$

The equation of the plane which contains the $y$-axis and passes through the point $(1, 2, 3)$ is:

If the line drawn from the point $(-2, -1, -3)$ meets a plane at a right angle at the point $(1, -3, 3)$,find the equation of the plane.

The equation of the plane passing through $(2, 3, 4)$ and parallel to the plane $5x - 6y + 7z = 3$ is:

Vedclass Test Series

Exam Paper Generator

Online Exam Module