Given, equation of curve is x2−xy+y2=27 .............(i) Taking derivative w.r.t.x on both sides 2x−
xdy
dx
−y+2y
dy
dx
=0 ⇒
dy
dx
(2y−x)=y−2x⇒
dy
dx
=
y−2x
2y−x
Since, curve has tangent parallel to X- axis. ∴ Slope of tangent =0 ⇒
dy
dx
=0⇒
y−2x
2y−x
=0 ⇒y=2x ..............(ii) Now, solving Eqs. (i) and (ii), we get x2−2x2+4x2=27⇒3x2=27⇒x=±3 For x=3,y=6 and x=−3,y=−6 ∴ Points are (3,6) and (−3,−6).