The equation of the plane that contains the l | vedclass

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

Question

(B) The equation of the family of planes passing through the intersection of the planes $x+2y+3z-4=0$ and $2x+y-z+5=0$ is given by $(x+2y+3z-4) + \lam

Vedclass · Accepted Answer

$33x+45y+50z-41=0$

Vedclass · Answer

(B) The equation of the family of planes passing through the intersection of the planes $x+2y+3z-4=0$ and $2x+y-z+5=0$ is given by $(x+2y+3z-4) + \lambda(2x+y-z+5) = 0$. Rearranging the terms,we get $(1+2\lambda)x + (2+\lambda)y + (3-\lambda)z + (-4+5\lambda) = 0$ ... $(1)$. Since this plane is perpendicular to the plane $5x+3y-6z+8=0$,the dot product of their normal vectors must be zero. Thus,$(1+2\lambda)(5) + (2+\lambda)(3) + (3-\lambda)(-6) = 0$. Expanding this,we get $5 + 10\lambda + 6 + 3\lambda - 18 + 6\lambda = 0$. Simplifying,$19\lambda - 7 = 0$,which gives $\lambda = \frac{7}{19}$. Substituting $\lambda = \frac{7}{19}$ into equation $(1)$: $(1 + 2(\frac{7}{19}))x + (2 + \frac{7}{19})y + (3 - \frac{7}{19})z + (-4 + 5(\frac{7}{19})) = 0$. $(\frac{19+14}{19})x + (\frac{38+7}{19})y + (\frac{57-7}{19})z + (\frac{-76+35}{19}) = 0$. $\frac{33}{19}x + \frac{45}{19}y + \frac{50}{19}z - \frac{41}{19} = 0$. Multiplying by $19$,we get $33x + 45y + 50z - 41 = 0$.

The equation of the plane that contains the line of intersection of the planes $x+2y+3z-4=0$ and $2x+y-z+5=0$ and is perpendicular to the plane $5x+3y-6z+8=0$ is

Similar Questions

Show that the lines $\frac{x+3}{-3}=\frac{y-1}{1}=\frac{z-5}{5}$ and $\frac{x+1}{-1}=\frac{y-2}{2}=\frac{z-5}{5}$ are coplanar.

The equation of the plane containing the lines $\frac{x - 1}{2} = \frac{y + 1}{\lambda} = \frac{z}{2}$ and $\frac{x + 1}{5} = \frac{y + 1}{2} = \frac{z}{\lambda}$ is

The equation of the plane containing the line $r = i + j + \lambda (2i + j + 4k)$ is

Find the length of the perpendicular from the point $(7, 14, 5)$ to the plane $2x + 4y - z = 2$ and the coordinates of the foot of the perpendicular.

$A$ symmetrical form of the line of intersection of the planes $x = ay + b$ and $z = cy + d$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module