Question

55. Pair of lines passing through (0,0) and perpendicular
to 6x2 - 2xy + 5y2 = 0 is
(a) 6x2 + 2xy + 5y2 = 0 (b) 5x² - 2xy + 6y2 = 0
(c) 6y + 2xy + 5x2 = 0 (d) 6y- 2xy - 5x2 = 0​

Answer 1

Answer:

Given homogeneous equation is 5x

2

+2xy−3y

2

=0 which is factorisable as

5x

2

+5xy−3xy−3y

2

=0

⇒5x(x+y)−3y(x+y)=0

⇒(x+y)(5x−3y)=0

x+y=0 and 5x−3y=0 are the two lines represented by the given equation.

⇒Their slopes are –1 and

3

5

.

Required two lines are respectively perpendicular to these lines.

∴ Slopes of required lines are 1 and −

5

3

and the lines pass through origin.

∴ Their individual equations are y=1⋅x and y=−

5

3

x

i.e.,x−y=0,3x+5y=0

∴ Their joint equation is (x−y)(3x+5y)=0

⇒3x

2

−3xy+5xy−5y

2

=0

⇒3x

2

+2xy−5y

2

=0

Hence 3x

2

+2xy−5y

2

=0 is the required joint equation.