The equation #x^2+y^2=c^2((bx+ay)/(ab))^2# can be expanded as
#a^2b^2x^2+a^2b^2y^2=c^2b^2x^2+c^2a^2y^2+2abc^2xy#
or #x^2(a^2b^2-b^2c^2)-2abc^2xy+y^2(a^2b^2-c^2a^2)=0#
or #b^2x^2(a^2-c^2)-2abc^2xy+a^2y^2(b^2-c^2)=0#
or #(b^2(a^2-c^2))/(a^2(b^2-c^2))x^2-(2abc^2)/(a^2(b^2-c^2))xy+y^2=0#
As there is no term containing #x#, #y# or constant term,
the two lines would be passing trough origin and lines could be #y=m_1x# and #y=m_2x#, and hence the quadratic equation would be
#(m_1x-y)(m_2x-y)=0# i.e. #m_1m_2x^2-(m_1+m_2)xy+y^2=0#
Hence comparing the two equations
#m_1m_2=(b^2(a^2-c^2))/(a^2(b^2-c^2))#
but as they form right angle #m_1m_2=-1#
i.e. #(b^2(a^2-c^2))/(a^2(b^2-c^2))=-1#
or #b^2a^2-b^2c^2=-a^2b^2+a^2c^2#
or #b^2a^2-b^2c^2+a^2b^2-a^2c^2=0#
or #b^2c^2+a^2c^2+a^2b^2=3a^2b^2#
and dividing by #a^2b^2c^2# we get
#1/a^2+1/b^2+1/c^2=3/c^2#
