Question

If the term containing xy' in the equation

x ^ 2 + 4xy + y ^ 2 - 2x + 2y - 6 = 0 is to be eliminated, the angle through which the axes should be rotated is
1) pi/2
2) pi/3
3) pi/4
4) pi​

Answer 1

Given data : -

Equation : x² + 4xy + y² - 2x + 2y - 6 = 0

To find :-

The angle through which the axes should be rotated ?

Solution :-

→ x² + 4xy + y² - 2x + 2y - 6 = 0

compare above equation with

ax² + 2hxy + by² + 2gx + 2fy + c = 0

Hence,

• a = 1

• 2h = 4 → h = 4/2 → h = 2

• b = 1

• 2g = - 2 → g = - 2/2 → g = - 1

• 2f = 2 → f = 2/2 → f = 1

• c = - 6

Let, θ be the angle if rotation of the axes

Now, {we use formula to find angle through which the axes should be rotated}

→ θ = ½ tan^( - 1 ) [ 2h/(a - b)]

→ θ = ½ tan^( - 1 ) [ 2×2/(1 - 1)]

→ θ = ½ tan^( - 1 ) [ 4/0]

→ θ = ½ tan^( - 1 ) [ ∞ ]

→ θ = ½ × π/2

→ θ = π/4

Answer :- 3) π/4 or pi/4

Hence, the π/4 is the angle through which the axes rotated.

More info :

Straight lines :

• Equation of the line passing through the point ( x1 , y1 ) and having slope m is y - y1 = m ( x - x1 )

Pair of lines :

• if θ is the acute angle between the lines ax² + 2hxy + by² = 0 or ax² + 2hxy + by² + 2gx + 2fy + c = 0,

Then,

→ tan θ = | {2√h²-ab}/{a+b} |

where, a + b ≠ 0

These lines are :

• coincident or paralle, if h² - ab = 0
• perpendicular to each other, if a + b = 0