Assertion: The points (2, 1, 5) and (3, 4, 3) | 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 $f(x, y, z) = 2x + 2y - 2z - 1 = 0$.To check if two points $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$ lie on opposit

Vedclass · Accepted Answer

$A$ and $R$ are both independently true and $R$ is the correct explanation for $A$.

Vedclass · Answer

(A) Let the equation of the plane be $f(x, y, z) = 2x + 2y - 2z - 1 = 0$.To check if two points $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$ lie on opposite sides of a plane,we evaluate $f(x_1, y_1, z_1)$ and $f(x_2, y_2, z_2)$. If the results have opposite signs,the points lie on opposite sides.For point $(2, 1, 5)$: $f(2, 1, 5) = 2(2) + 2(1) - 2(5) - 1 = 4 + 2 - 10 - 1 = -5$.For point $(3, 4, 3)$: $f(3, 4, 3) = 2(3) + 2(4) - 2(3) - 1 = 6 + 8 - 6 - 1 = 7$.Since $f(2, 1, 5) = -5$ and $f(3, 4, 3) = 7$ have opposite signs,the points lie on opposite sides of the plane.The algebraic perpendicular distance is given by $\frac{f(x, y, z)}{\sqrt{a^2 + b^2 + c^2}}$. Since the denominator is always positive,the sign of the distance depends solely on the sign of $f(x, y, z)$.Thus,both the Assertion and the Reason are true,and the Reason correctly explains the Assertion.

Assertion: The points $(2, 1, 5)$ and $(3, 4, 3)$ lie on opposite sides of the plane $2x + 2y - 2z - 1 = 0$.
Reason: The algebraic perpendicular distances from the given points to the plane have opposite signs.

Similar Questions

If the foot of the perpendicular drawn from $(0,0,0)$ to a plane is $(1,2,3)$,then the equation of the plane is:

The perpendicular distance from the origin to the plane containing the points having position vectors $\hat{i}+2\hat{j}+3\hat{k}$, $2\hat{i}+3\hat{j}-4\hat{k}$, and $3\hat{i}-4\hat{j}+5\hat{k}$ is

The locus represented by $xy + yz = 0$ is

The vector equation of the plane passing through the point $A(1, 2, -1)$ and parallel to the vectors $2 \hat{i} + \hat{j} - \hat{k}$ and $\hat{i} - \hat{j} + 3 \hat{k}$ is

If $(2,3,-1)$ is the foot of the perpendicular from $(4,2,1)$ to a plane,then the equation of the plane is

Vedclass Test Series

Exam Paper Generator

Online Exam Module

Assertion: The points $(2, 1, 5)$ and $(3, 4, 3)$ lie on opposite sides of the plane $2x + 2y - 2z - 1 = 0$.Reason: The algebraic perpendicular distances from the given points to the plane have opposite signs.