The three planes x + y = 0, y + z = 0 and x + z = 0

Question

The three planes x + y = 0, y + z = 0 and x + z = 0 (A) meet in a unique point (B) meet in a line (C) meet taken two at a time in parallel lines (D) None of these

Tardigrade Admin · Accepted Answer

The planes x + y = 0 i.e. x = - y and y + z = 0 i.e. z = -y meet in the line

\frac{x}{1} = \frac{y}{- 1} = \frac{z}{1}

.
Any point on this line is (t, -t, t). This point lies in the plane x + z = 0 if t + t = 0

\Rightarrow

t = 0 .
So the three planes meet in a unique point (0, 0, 0).

Q. The three planes x + y = 0, y + z = 0 and x + z = 0

meet in a unique point

meet in a line

meet taken two at a time in parallel lines

None of these

The planes x + y = 0 i.e. x = - y and y + z = 0 i.e. z = -y meet in the line 1x​=−1y​=1z​ . Any point on this line is (t, -t, t). This point lies in the plane x + z = 0 if t + t = 0 ⇒ t = 0 . So the three planes meet in a unique point (0, 0, 0).

The planes x + y = 0 i.e. x = - y and y + z = 0 i.e. z = -y meet in the line
$\frac{x}{1} = \frac{y}{- 1} = \frac{z}{1}$ .
Any point on this line is (t, -t, t). This point lies in the plane x + z = 0 if t + t = 0 $\Rightarrow$ t = 0 .
So the three planes meet in a unique point (0, 0, 0).