We have, 3x2+4y2−xy+k=0 is transformed equation after shifting origin to the point ( α,β ) replacing x⟶x−α and y⟶y−β ∴3(x−α)2+4(y−β)2−(x−α)(y−β)+k=0 ⇒3(x2+α2−2αx)+4(y2+β2−2βy) −xy+αy+βx−αβ+k=0 ⇒3x2+4y2−xy+x(−6α+β)+y(−8β+α) +3α2+4β2−αβ+k=0 Comparing Eq. (i) with given equation, 3x2+4y2−xy−5x−7y+2=0, we get −6α+β=−5 α−8β=−7 3α2+4β2−αβ+k=2 On solving Eqs. (ii) and (iii), we get α=β=1 By Eqs. (ii) 3+4−1+k=2 ⇒k=−4 ∴α+β−k=1+1+4=6.
