Find the point to which the origin should be shifted after a translation of axes so that the following equations will have no first degree term: `y^2+x^2-4x-8y+3=0`

Explanation

Substituting `x` and `y` with `(x + p)` and `(y + q)` respectively, we have

`= (x + p)^2 + (y + q)^2 – 4(x + p) - 8(y + q) + 3 = 0`

`= x^2 + p^2 + 2px - y^2 – q^2 – 2qy – 4x – 4p – 8y – 8q + 3 = 0 `

`=x^2 + y^2 + x(2p – 4) + y(2q – 8) + p^2 + q^2 – 4p – 8q + 3 = 0`
For first degree term to be zero we have, `2p – 4 = 0` and `2q – 8 = 0`

Giving us, `p = 2` and `q = 4`.

Hence, the shifted point is `(p, q) = (2, 4)`.