Question

A variable plane passes through a fixed point $a,b,c$ and meets the co-ordinate axes in $A,B,C$. The locus of the point common to plane through $A,B,C$ parallel to coordinates planes is

• A. $ayz+bza+cxy=xyz$
• B. $axy+byz+czx=xyz$
• C. $axy+byz+czx=abc$
• D. $bcx+acy+abz=abc$

Answer 1

Answer: (A)

$ayz+bza+cxy=xyz$

Let the plane be $\frac{x}{\alpha }+\frac{y}{\beta }+\frac{z}{\gamma }=1$
If passes through $(a,b,c)$
$\therefore \frac{a}{\alpha }+\frac{b}{\beta }+\frac{c}{\gamma }=1$ ...$\left(1\right)$
Now coordinate of point $A,B,C$ are $(\alpha ,0,0),(0,\beta ,0)$ and $(0,0,\gamma )$ respectively.
Equation of the planes through $A,B,C$ parallel to coordinate plane are
$x=\alpha$ ...$\left(2\right)$
$y=\beta$ ...$\left(3\right)$
$z=\gamma$ ...$\left(4\right)$
The locus of this point of intersection will be obtained by eliminating $\alpha ,\beta ,\gamma$ from the these with the help of relation $\left(1\right)$.
Thus, we get $\frac{a}{x}+\frac{b}{y}+\frac{c}{z}=1$
i.e $ayz+bxz+cxy=xyz$