Question

When the axes are rotated through an angle 45°, the transformed equation of a curve is 17x² – 16xy + 17y² = 225. Find the original equation of the curve.

Answer 1

Angle of rotation θ = 45°

x = x cos θ + y sin θ

= 2cos 45° + ysin45°

y = –x sinθ + ycosθ

= – xsin45° + ycos45°

The original equation of 17x² – 16xy + 17y² = 225 is

⇒ 17x² + 34xy + 17y² – 16y² + 16x² + 17x² – 34xy + 17y² = 450 

⇒ 50x² + 18y² = 450

⇒ 25x² + 9y² = 225