Question

If `4x^2-12xy+Ay^2=0` represents a pair of perpendicular lines through origin, find `A`?

Answer 1

`A=-4`

Explanation:

As we have two perpendicular lines equations must be of the type

`ax+by=0` and `bx-ay=0`, so that multiplicaton of their slopes is `-1`

Hence equation should be `(ax+by)(bx-ay)=0`

or `abx^2-aby^2+(b^2-a^2)xy=0`

i.e. coefficients of `x^2` and `y^2` should be equal but opposite in size

Hence `A=-4`

and then our equation is `4x^2-12xy-4y^2=0`

and graph appears as shown below:

graph{4x^2-4y^2-12xy=0 [-10, 10, -5, 5]}