Assertion: The points (2, 1, 5) and (3, 4, 3) | vedclass

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(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

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

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(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.

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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.