Prove that the straight lines joining the origin to the points of inte

1.	Prove that the straight lines joining the origin to the points of intersection of 3x2 + 5xy – 3y2 + 2x + 3y = 0 and 3x – 2y – 1 = 0 are at right angles.
Answer» Homogenizing the given equations 3x² + 5xy – 3y² + 2x + 3y = 0 and 3x – 2y – 1 = 0 (i.e) 3x – 2y = 1. We get (3x² + 5xy – 3y²) + (2x + 3y)( 1) = 0 (i.e) (3x² + 5xy – 3y²) + (2x + 3y)(3x – 2y) = 0 3x² + 5xy – 3y² + bx² – 4xy + 9xy – 6y² = 0 9x² + 10xy – 9y² = 0 Coefficient of x² + coefficient of y² = 9 – 9 = 0 ⇒ The pair of straight lines are at right angles.

Prove that the straight lines joining the origin to the points of intersection of 3x2 + 5xy – 3y2 + 2x + 3y = 0 and 3x – 2y – 1 = 0 are at right angles.

Answer»

Homogenizing the given equations 3x² + 5xy – 3y² + 2x + 3y = 0 and 3x – 2y – 1 = 0

(i.e) 3x – 2y = 1.

We get (3x² + 5xy – 3y²) + (2x + 3y)( 1) = 0

(i.e) (3x² + 5xy – 3y²) + (2x + 3y)(3x – 2y) = 0

3x² + 5xy – 3y² + bx² – 4xy + 9xy – 6y² = 0

9x² + 10xy – 9y² = 0

Coefficient of x² + coefficient of y² = 9 – 9 = 0

⇒ The pair of straight lines are at right angles.