A plane is parallel to two lines,whose direct | vedclass

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

Question

(C) The equation of a plane passing through $(1, 1, 1)$ is given by $a(x-1) + b(y-1) + c(z-1) = 0$. Since the plane is parallel to the lines with dire

Vedclass · Accepted Answer

$\frac{9}{2}$

Vedclass · Answer

(C) The equation of a plane passing through $(1, 1, 1)$ is given by $a(x-1) + b(y-1) + c(z-1) = 0$. Since the plane is parallel to the lines with direction ratios $(1, 0, -1)$ and $(-1, 1, 0)$,the normal vector $(a, b, c)$ must be perpendicular to both direction vectors. Thus,$a(1) + b(0) + c(-1) = 0 \Rightarrow a - c = 0$ and $a(-1) + b(1) + c(0) = 0 \Rightarrow -a + b = 0$. This implies $a = c$ and $a = b$,so $a = b = c$. Substituting $a=b=c$ into the plane equation,we get $a(x-1) + a(y-1) + a(z-1) = 0$,which simplifies to $x + y + z = 3$. Dividing by $3$,we get the intercept form $\frac{x}{3} + \frac{y}{3} + \frac{z}{3} = 1$. Thus,the coordinates of $A, B, C$ are $(3, 0, 0), (0, 3, 0),$ and $(0, 0, 3)$ respectively. The volume of the tetrahedron $OABC$ is given by $V = \frac{1}{6} |x_A y_B z_C| = \frac{1}{6} |3 	imes 3 	imes 3| = \frac{27}{6} = \frac{9}{2}$ cubic units.

$A$ plane is parallel to two lines,whose direction ratios are $1, 0, -1$ and $-1, 1, 0$ and it contains the point $(1, 1, 1)$. If it cuts coordinate axes ($X, Y, Z$-axes respectively) at $A, B, C$,then the volume of the tetrahedron $OABC$ is (in cubic units):

Similar Questions

$O$ is the origin and $A$ is $(a, b, c)$. Find the direction cosines of the line $OA$ and the equation of the plane passing through $A$ at a right angle to $OA$.

Find the equation of the set of points which are equidistant from the points $A(3, 4, -5)$ and $B(-2, 1, 4)$.

The locus of a first degree equation in $x, y, z$ is a

Let $P$ be a plane passing through the points $(2,1,0)$,$(4,1,1)$,and $(5,0,1)$,and $R$ be the point $(2,1,6)$. Then the image of $R$ in the plane $P$ is:

The equation of the plane passing through the points $(1, 2, -3)$ and $(2, -2, 1)$ and parallel to the $X$-axis is:

Vedclass Test Series

Exam Paper Generator

Online Exam Module