A variable plane passes through the fixed point $(3,2,1)$ and meets $X, Y$ and $Z$ axes at points $A$, B and…

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
  1. $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{11}{6}$
  2. $\frac{x}{3}+\frac{y}{2}+\frac{z}{1}=1$
  3. $\frac{3}{x}+\frac{2}{y}+\frac{1}{z}=1$
  4. $x+y+z=6$

Solution

Let the plane be $\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1$ If passes through $(3,2,1)$ $\therefore \quad \frac{3}{a}+\frac{2}{b}+\frac{1}{c}=1$ Now, coordinates of points A, B, C are (a, 0, 0), $(0, b, 0)$ and $(0,0, c)$ respectively. $\therefore \quad$ Equations of the planes passing through $\mathrm{A}, \mathrm{B}, \mathrm{C}$ are $x=\mathrm{a}, y=\mathrm{b}$ and $\mathrm{z}=\mathrm{c}$ respectively. $\therefore \quad$ From equation (i), we get Required locus is $\frac{3}{x}+\frac{2}{y}+\frac{1}{z}=1$

Asked in: MHT CET 2024 (11 May Shift 2)

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