The equation of the plane passing through the | vedclass

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

Question

(C) The equation of any plane passing through the intersection of the planes $P_1: x+y+z-1=0$ and $P_2: 2x+3y-z+4=0$ is given by $P_1 + \lambda P_2 =

Vedclass · Accepted Answer

$y-3z+6=0$

Vedclass · Answer

(C) The equation of any plane passing through the intersection of the planes $P_1: x+y+z-1=0$ and $P_2: 2x+3y-z+4=0$ is given by $P_1 + \lambda P_2 = 0$. $(x+y+z-1) + \lambda(2x+3y-z+4) = 0$ $(1+2\lambda)x + (1+3\lambda)y + (1-\lambda)z + (4\lambda-1) = 0$. Since the plane is parallel to the $x$-axis, its normal vector $\vec{n} = (1+2\lambda, 1+3\lambda, 1-\lambda)$ must be perpendicular to the $x$-axis vector $\hat{i} = (1, 0, 0)$. Therefore, the dot product is zero: $(1+2\lambda)(1) + (1+3\lambda)(0) + (1-\lambda)(0) = 0$. $1+2\lambda = 0 \Rightarrow \lambda = -\frac{1}{2}$. Substituting $\lambda = -\frac{1}{2}$ into the equation: $(x+y+z-1) - \frac{1}{2}(2x+3y-z+4) = 0$. Multiplying by $2$: $2x+2y+2z-2 - 2x-3y+z-4 = 0$. $-y + 3z - 6 = 0$, which simplifies to $y-3z+6=0$.

The equation of the plane passing through the intersection of the planes $x+y+z=1$ and $2x+3y-z+4=0$ and parallel to the $x$-axis is:

Similar Questions

On which of the following lines does the point of intersection of the line $\frac{x-4}{2}=\frac{y-5}{2}=\frac{z-3}{1}$ and the plane $x+y+z=2$ lie?

If the three planes $x = 5, 2x - 5ay + 3z - 2 = 0$ and $3bx + y - 3z = 0$ contain a common line,then $(a, b)$ is equal to

If the equation of a line is $\frac{x + 3}{2} = \frac{y - 4}{3} = \frac{z + 5}{2}$ and the equation of a plane is $4x - 2y - z = 1$,then which of the following is true?

The line $\frac{x-2}{3}=\frac{y-3}{4}=\frac{z-4}{5}$ is parallel to the plane

The equation of the plane passing through the line $\frac{x-1}{2} = \frac{y+1}{-1} = \frac{z-3}{4}$ and perpendicular to the plane $x+2y+z=12$ is given by $ax+by+cz+4=0$. Then:

Vedclass Test Series

Exam Paper Generator

Online Exam Module