A variable plane passes through the fixed point ( 3 , 2 , 1 ) and meets X , Y and Z axes at points A , B and C respectively. A plane is drawn parallel to YZ - plane through A , a second plane is drawn

Question

A variable plane passes through the fixed point ( 3 , 2 , 1 ) and meets X , Y and Z axes at points A , B and C respectively. A plane is drawn parallel to YZ - plane through A , a second plane is drawn parallel to ZX -plan through B, a third plane is drawn parallel to XY - plane through C . Then locus of the point of intersection of these three planes, is

TestHub · Accepted Answer

Let the equation of the variable plane be $\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1$ .

Since the plane passes through the fixed point $(3, 2, 1)$ , we have:

$\frac{3}{a} + \frac{2}{b} + \frac{1}{c} = 1$

The plane meets the coordinate axes at the points $A (a, 0, 0)$ , $B (0, b, 0)$ , and $C (0, 0, c)$ .

The equation of the plane passing through $A$ and parallel to the YZ-plane is $x = a$ .

The equation of the plane passing through $B$ and parallel to the ZX-plane is $y = b$ .

The equation of the plane passing through $C$ and parallel to the XY-plane is $z = c$ .

Let the point of intersection of these three planes be $(x, y, z)$ . Then we have $x = a$ , $y = b$ , and $z = c$ .

Substituting these values into the condition $\frac{3}{a} + \frac{2}{b} + \frac{1}{c} = 1$ , we get the locus of the point of intersection as:

$\frac{3}{x} + \frac{2}{y} + \frac{1}{z} = 1$

Answer: $\frac{3}{x} + \frac{2}{y} + \frac{1}{z} = 1$

Mathematics - 3D Coordinate Geometry Question with Solution | TestHub

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