A point `P` moves on a plane `x/a+y/b+z/c=1.` A plane through `P` and perpendicular to `O P` meets the coordinate axes at `A , Ba n d Cdot` If the planes through `A ,Ba n dC` parallel to the planes `x=0,y=0a n dz=0,` respectively, intersect at `Q ,` find the locus of `Qdot`

Answer

`(1)/(ax)+ (1)/(by)+(1)/(cz)= (1)/(x^(2))+(1)/(y^(2))+ (1)/(z^(2))`

Explanation

The given plane is `(x)/(a)+(y)/(b)+(z)/(c)=1" "(i)`
Let P(h,k,l) be the point on the plane. Then
`(h)/(a)+(k)/(b)+(l)/(c)=1" "(ii)`
`impliesOPsqrt(h^(2)+k^(2)+l^(2))`
Direction cosines of OP are
`(h)/sqrt(h^(2)+k^(2)+l^(2)),(k)/sqrt(h^(2)+k^(2)+l^(2))` and `(h)/sqrt(h^(2)+k^(2)+l^(2))`
The equation of the plane through P and normal to OP is
`(hx)/sqrt(h^(2)+k^(2)+l^(2))+(ky)/sqrt(h^(2)+k^(2)+l^(2))+(lz)/sqrt(h^(2)+k^(2)+l^(2))`
`=sqrt(h^(2)+k^(2)+l^(2))`
or `hx+ky+lz=h^(2)+k^(2)+l^(2)`
Therefore,
`A-=((h^(2)+k^(2)+l^(2))/(h),0,0)`,
`B-=(0,(h^(2)+k^(2)+l^(2))/(k),0)`
and `C-=(0,0,(h^(2)+k^(2)+l^(2))/(l))`
If Q `(alpha,beta,gamma)`, then
`alpha=(h^(2)+k^(2)+l^(2))/(h),beta=(h^(2)+k^(2)+l^(2))/(k)`
and `gamma=(h^(2)+k^(2)+l^(2))/(l)" "(iii)`
Now, `(1)/(a^(2))+(1)/(beta^(2))+(1)/(gamma^(2))`
`=(h^(2)+k^(2)+l^(2))/((h^(2)+k^(2)+l^(2))^(2))=(1)/(h^(2)+k^(2)+l^(2))" "(iv)`
From (iii), `h=(h^(2)+k^(2)+l^(2))/(alpha)or(h)/(a)=(h^(2)+k^(2)+l^(2))/(aalpha)`
Similarly, `(k)/(b)=(h^(2)+k^(2)+l^(2))/(b beta)and(1)/(c)=(h^(2)+k^(2)+l^(2))/(cgamma)`
`(h^(2)+k^(2)+l^(2))/(aalpha)+(h^(2)+k^(2)+l^(1))/(b beta)+(h^(2)+k^(2)+l^(2))/(cgamma)`
`=(h)/(a)+(k)/(b)+(l)/(c)=1" "["from"(ii)]`
or `(1)/(aalpha)+(1)/(b beta)+(1)/(cgamma)=(1)/(h^(2)+k^(2)+l^(2))=(1)/(alpha)+(1)/(beta^(2))+(1)/(gamma^(2))`
[from (iv)]
The required equation of locus is
`(1)/(ax)+(1)/(by)+(1)/(cz)+(1)/(x^(2))+(1)/(y^(2))+(1)/(z^(2)`