Question

11.
Show that by shifting the origin to a suitable point, the equation
6x2 + 5 x y-6 y2 - 17 x + 7 y + 5 = 0 may be transformed to
6x² + 5xy - 6y2 = 0.​

Answer 1

Solution :

Let the origin be shifted to the point (a, b) and the transformation formulae are

x = x' + a , y = y' + b

Then, the given equation becomes

6 (x' + a)² + 5 (x' + a) (y' + b) - 6 (y' + b)² - 17 (x' + a) + 7 (y' + b) + 5 = 0

or, 6 (x'² + 2ax' + a²) + 5 (x'y' + bx' + ay' + ab) - 6 (y'² + 2by' + b'²) - 17 (x' + a) + 7 (y' + b) + 5 = 0

or, 6x'² + 5x'y' - 6y'² + (12a + 5b - 17)x' + (5a - 12b + 7)y' + (6a² + 5ab - 6b² - 17a + 7b + 5) = 0

The coefficients of x' and y' in the transformed equation are

(12a + 5b - 17) and (5a - 12b + 7),

which have to be separately zero, if the first degree terms are to be removed.

Thus 12a + 5b - 17 = 0 and 5a - 12b + 7 = 0

Solving, we get a = b = 1.

Therefore, the origin must be shifted to the point (1, 1).

Hence, we conclude that the given equation can be transformed to the required form with suitable shifting of the origin to (1, 1).

This completes the proof.