Question

The Cartesian equation of a plane which passes through the points $\text{A}(2, 2, 2)$ and making equal nonzero intercepts on the co-ordinate axes is

• A. $x + y + z = 6$
• B. $x - 2y + z = 0$
• C. $2x + y + z = 7$
• D. $x - y + z = 2$

Hint:

Use a quick point substitution check to save time! Plug the coordinates $(2,2,2)$ straight into the options:
For (A): $2 + 2 + 2 = 6$ (True)
For (B): $2 - 4 + 2 = 0$ (True, but does not have equal non-zero intercepts)
Checking options using point validation is an excellent way to confirm your answer!

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
We need to find the Cartesian equation of a plane that passes through the coordinates $(2,2,2)$ and makes completely identical, non-zero intercept lengths on the $x$, $y$, and $z$ axes.

Step 2: Key Formula or Approach:
The intercept form of a plane equation in three-dimensional space is given by:
$$\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1$$ Where $a$, $b$, and $c$ are the intercepts on the $x$, $y$, and $z$ axes respectively. Since the problem specifies that all three intercepts are identical, we can set $a = b = c$, simplifying the equation to:
$$\frac{x}{a} + \frac{y}{a} + \frac{z}{a} = 1 \implies x + y + z = a$$

Step 3: Detailed Explanation:
We are given that the plane passes through the point $\text{A}(2, 2, 2)$. Therefore, substituting these coordinates into our simplified plane equation must satisfy the equality:
$$2 + 2 + 2 = a \implies a = 6$$ Since the intercept parameter $a$ is found to be 6, substitute it back into the equation to get the final Cartesian form:
$$x + y + z = 6$$ This matches option (A).

Step 4: Final Answer:
The Cartesian equation of the plane is $x + y + z = 6$, which corresponds to option (A).