Question

If the lines represented by the equation `x^2+y^2=c^2((bx+ay)/(ab))^2` form a right angle then prove that:`1/a^2+1/b^2+1/c^2=3/c^2`?

`1/a^2+1/b^2+1/c^2=3/c^2`

Answer 1

See below.

Explanation:

The product of lines is homogeneous in `x,y` so the lines should be of the form

`L_1 -> y-m x = 0`
`L_2->x+my = 0`

then

`(a^2 b^2 - b^2 c^2) x^2 - 2 a b c^2 x y + (a^2 b^2 - a^2 c^2) y^2 = (y-mx)(x+my)`

grouping variables gives us

#{(a^2 b^2 - a^2 c^2 - m = 0), ( m^2- 2 a b c^2 =1), (a^2 b^2 - b^2 c^2 + m=0):}#

From the first we have

`1/c^2-1/b^2 = m/(a^2b^2c^2)`

from the third we have

`1/c^2-1/a^2 = -m/(a^2b^2c^2)`

adding those last equations we have

`2/c^2-1/a^2-1/b^2= 0` now adding `1/c^2` to both sides

`3/c^2-1/a^2-1/b^2=1/c^2` or

`1/a^2+1/b^2+1/c^2=3/c^2`