What is the differential equation corresponding to `y^2 -2ay+x^2 = a^2` by eliminating a?

Options

A. ` (x^(2) -2y^(2))p^(2)-4pxy -x^(2)=0`
B. `(x^(2)-2y^(2))p^(2)+ pxy -x^(2)=0`
C. `(x^(2)+2y^(2))p^(2)- pxy -x^(2)=0`
D. `(x^(2)+2y^(2))p^(2)- pxy +x^(2)=0`

Answer

A

Explanation

`y^(2) -2 (y) (a) =a^(2) =x^(2) = a^(2) +a^(2)`
`Rightarrow (y-a)^(2) +x^(2) =2a^(2)`
Diff w.r.t x
` 2(y -a) .y^(1) +2x=0`
` Rightarrow y-a = (-x)/p`
` Rightarrow a= y + x/p`
(i) ` Rightarrow ((-x)/p)^(2) +x^(2) =2 (y + x/p)^(2)`
` 2 (((py +x)^(2))/(p2))`
`(x^(2)+p^(2)x^(2))/(cancel(p^(2)))=(2(p^(2)y^(2)+x^(2)+2pxy))/(cancel(p^(2)))`
` Rightarrow x^(2) +p^(2)x^(2)-2p^(2)y^(2) -2x^(2) -4pxy =0`
` Rightarrow (x^(2) -2y^(2) p^(2)-4pxy =0`