A plane passing through a point (2,2,2) cuts | vedclass

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

Question

(A) Let the equation of the plane be $\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1$.Since the plane passes through $(2, 2, 2)$,we have $\frac{2}{a} + \

Vedclass · Accepted Answer

$\alpha + \beta + \gamma \ge \frac{9}{2}$

Vedclass · Answer

(A) Let the equation of the plane be $\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1$.Since the plane passes through $(2, 2, 2)$,we have $\frac{2}{a} + \frac{2}{b} + \frac{2}{c} = 1$,which implies $\frac{1}{a} + \frac{1}{b} + \frac{1}{c} = \frac{1}{2}$.The coordinates of $A, B, C$ are $(a, 0, 0), (0, b, 0), (0, 0, c)$ respectively.The centroid $P(\alpha, \beta, \gamma)$ of tetrahedron $OABC$ is given by $\alpha = \frac{0+a+0+0}{4} = \frac{a}{4}$,$\beta = \frac{b}{4}$,and $\gamma = \frac{c}{4}$.Thus,$a = 4\alpha, b = 4\beta, c = 4\gamma$.Substituting these into the plane equation: $\frac{1}{4\alpha} + \frac{1}{4\beta} + \frac{1}{4\gamma} = \frac{1}{2} \Rightarrow \frac{1}{\alpha} + \frac{1}{\beta} + \frac{1}{\gamma} = 2$.Using the $A.M. \ge H.M.$ inequality for positive numbers $\alpha, \beta, \gamma$:$\frac{\alpha + \beta + \gamma}{3} \ge \frac{3}{\frac{1}{\alpha} + \frac{1}{\beta} + \frac{1}{\gamma}} = \frac{3}{2}$.Therefore,$\alpha + \beta + \gamma \ge \frac{9}{2}$.

$A$ plane passing through a point $(2,2,2)$ cuts the positive semi-axes at $A$,$B$,and $C$. If $P(\alpha, \beta, \gamma)$ is the centroid of the tetrahedron $OABC$ (where $O$ is the origin),then select the correct option.

Similar Questions

$A$ plane passes through the point $(3, 5, 7)$. If the direction ratios of its normal are equal to the intercepts made by the plane $x+3y+2z=9$ with the coordinate axes,then the equation of that plane is

The length of the perpendicular drawn from the point $(2, 1, 4)$ to the plane containing the lines $\vec r = (\hat i + \hat j) + \lambda (\hat i + 2\hat j - \hat k)$ and $\vec r = (\hat i + \hat j) + \mu (-\hat i + \hat j - 2\hat k)$ is

The vertices of a tetrahedron are $O(0, 0, 0)$,$A(1, 2, 1)$,$B(2, 1, 3)$,and $C(-1, 1, 2)$. Find the angle between the faces $OAB$ and $ABC$.

The equation of the plane which is parallel to the $xy$-plane and cuts an intercept of length $3$ from the $z$-axis is

If the foot of the perpendicular drawn from the origin to a plane is $(1, 2, 3)$,then a point on that plane is

Vedclass Test Series

Exam Paper Generator

Online Exam Module