Question

Find the angle through which the
axes are to be rotated so as to rem-
ove the xy term in the equationx2 4xy + y2 - 2x + 2y - 6 = 0.​

Answer 1

Answer:

Explanation:

Given that,

New co-ordinates are (3, 4)

X = 3, Y = 4

x = X cos θ – Y sin θ

= 3 cos60 – 4 sin60

= 3 (½) – 4 3√2,

= 3−43√2.

y = X sin θ + Y cos θ

= 3 sin60 + 4 cos60

= 3 3√2 + 4 (½)

= 4+3√2.

Co-ordinate of P are (3−43√2,4+3√2).

2) Find the angle through which the axes are to be rotated so as to remove the xy term in the equation x² + 4xy + y² – 2x + 2y – 6 = 0.

Solution: Comparing the equation

x² + 4xy + y² – 2x + 2y – 6 = 0 with ax² + 2hxy + by² + 2gx + 2fy + c = 0

a = 1, h = 2, b = 1, g = -1, f = 1, c = -6

let θ be the angle of rotation of axes, then θ = ½ tan⁻¹(2ha−b),

= ½ tan⁻¹ (4/ 1 – 1) = ½ tan⁻¹ (4/0)

= ½ tan⁻¹(∞) = ½ x π/2

θ = π/4