Question

A variable plane passes through a fixed point `(alpha,beta,gamma)` and meets the axes at `A ,B ,a n dCdot` show that the locus of the point of intersection of the planes through `A ,Ba n dC` parallel to the coordinate planes is `alphax^(-1)+betay^(-1)+gammaz^(-1)=1.`

Answer 1

Let the equation of the variable plane be `(x)/(a)+(y)/(b)+(z)/(c)=1" "(i)`
where a, b and c are the parameters.
Plane (i) passes through the piont `(alpha,beta,gamma)`. Therefore, `(alpha)/(a)+(beta)/(b)+(gamma)/(c)=1" "(ii)`
Plane (i) meets the coordinate axes at points A, B and C. The equations of the planes passing through A, B and C and parallel to the coordinate planes are, respectively.
`x=a,y=b,z=c" "(iii)`
The locus of the point of intersection of these planes is obtained by eliminating the parameters a, b and c between (ii) and (iii). Putting the values of a, b and c from (iii) in (ii), the required locus is given by
`(alpha)/(a)+(beta)/(b)+(gamma)/(c)=1oralphax^(-1)+betay^(-1)+gammaz^(-1)=1`