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#

1 Answer
Dec 10, 2017

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#