Question

Transforming to parallel axes through a point (p, q), the equation x² + 3xy + 4y² + x + 18y + 25 = 0 becomes 2x² + 3xy + 4y² = 0, then ______.

Options

• p = −2, q = 3

• p = 2, q = −3

• p = 3, q = −4

• p = −4, q = 3

Answer 1

Transforming to parallel axes through a point (p, q), the equation x² + 3xy + 4y² + x + 18y + 25 = 0 becomes 2x² + 3xy + 4y² = 0, then p = 2, q = −3.

Explanation:

2x² + 3xy + 4y² = 0

i.e. 2X² + 3XY + 4Y² = 0,

Replacing X by x − p and Y by y − q, we get

2(x − p)² + 3(x − p)(y − q) + 4(y − q)² = 0

⇒ 2(x² − 2xp + p²) + 3(xy − xq − py + pq) + 4(y² − 2qy + q²) = 0

⇒ 2x² + 3xy + 4y² − x(4p + 3q) − y(3p + 8q) + 2p² + 3pq + 4q² = 0

Comparing the above equation with

2x² + 3xy + 4y² + x + 18y + 25 = 0, we get

4p + 3q = −1    ...(i)

3p + 8q = −18     ...(ii)

On solving (i) and (ii), we get

p = 2, q = −3