If the equation of the plane passing through | vedclass

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

Question

(A) The equation of the family of planes passing through the intersection of the planes $P_1: 2x - y + z - 3 = 0$ and $P_2: 4x - 3y + 5z + 9 = 0$ is g

Vedclass · Accepted Answer

$14$

Vedclass · Answer

(A) The equation of the family of planes passing through the intersection of the planes $P_1: 2x - y + z - 3 = 0$ and $P_2: 4x - 3y + 5z + 9 = 0$ is given by $(2x - y + z - 3) + \lambda(4x - 3y + 5z + 9) = 0$.Rearranging the terms,we get $x(2 + 4\lambda) + y(-1 - 3\lambda) + z(1 + 5\lambda) + (-3 + 9\lambda) = 0$.Since this plane is parallel to the line with direction ratios $(-2, 4, 5)$,the normal vector of the plane must be perpendicular to the line. Thus,the dot product of the normal vector $(2 + 4\lambda, -1 - 3\lambda, 1 + 5\lambda)$ and the direction vector $(-2, 4, 5)$ must be zero:$-2(2 + 4\lambda) + 4(-1 - 3\lambda) + 5(1 + 5\lambda) = 0$.$-4 - 8\lambda - 4 - 12\lambda + 5 + 25\lambda = 0$.$5\lambda - 3 = 0 \implies \lambda = \frac{3}{5}$.Substituting $\lambda = \frac{3}{5}$ into the plane equation:$(2x - y + z - 3) + \frac{3}{5}(4x - 3y + 5z + 9) = 0$.$5(2x - y + z - 3) + 3(4x - 3y + 5z + 9) = 0$.$10x - 5y + 5z - 15 + 12x - 9y + 15z + 27 = 0$.$22x - 14y + 20z + 12 = 0$.Dividing by $2$,we get $11x - 7y + 10z + 6 = 0$.Comparing this with $ax + by + cz + 6 = 0$,we get $a = 11, b = -7, c = 10$.Therefore,$a + b + c = 11 - 7 + 10 = 14$.

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 $.............$.

Similar Questions

The plane $3x + 4y + 6z + 7 = 0$ is rotated about the line $r = (\hat{i} + 2\hat{j} - 3\hat{k}) + t(2\hat{i} - 3\hat{j} + \hat{k})$ until the plane passes through the origin. The equation of the plane in the new position is

If the equation of the plane passing through the line of intersection of the planes $2x - 7y + 4z - 3 = 0$ and $3x - 5y + 4z + 11 = 0$ and the point $(-2, 1, 3)$ is $ax + by + cz - 7 = 0$,then the value of $2a + b + c - 7$ is

If $\theta$ is the angle between the line $\frac{x + 1}{3} = \frac{y - 2}{2} = \frac{z - 2}{4}$ and the plane $2x + y - 3z + 4 = 0$,then $64 \csc^2 \theta$ is equal to

The distance of the point $O(\vec{0})$ from the plane $\vec{r} \cdot(\hat{i}+\hat{j}+\hat{k})=5$ measured parallel to the vector $2 \hat{i}+3 \hat{j}-6 \hat{k}$ is:

$A$ straight line joining the points $(1, 1, 1)$ and $(0, 0, 0)$ intersects the plane $2x + 2y + z = 10$ at

Vedclass Test Series

Exam Paper Generator

Online Exam Module