Find the condition that one of the straight lines given by the equation ax^2+2hxy+by^2=0 may coincide with one of those given by the equation a_1x^2+2h_1xy+b_1y^2?

1 Answer
Aug 6, 2018

Given equations of pair of straight lines are

ax^2+2hxy+by^2=0......(1)

and

a_1x^2+2h_1xy+b_1y^2=0......(2)
Obviously the straight lines are passing through orin as suggested by the equations.

Now let color(red)(y=mx....(3)) be the equation of the line which coincides.

So by (1) and (3) we get

ax^2+2hx*mx+bm^2x^2=0

color(blue)(=>bm^2+2hm+a=0......(4))

Similarly by (2) and (3) we will get

a_1x^2+2h_1x*mx+b_1m^2x^2=0

color(magenta)(=>b_1m^2+2h_1m+a_1=0......(5))

By cross multiplication of (4) and (5) we get

m^2/(2ha_1-2h_1a)=m/(ab_1-a_1b)=1/(2h_1b-2hb_1)

Hence we get two values of m

(1) m=(2(ha_1-h_1a))/(ab_1-a_1b)
and
(2) m= (ab_1-a_1b)/(2(h_1b-hb_1))

By the condition of the problem these two values of m must be same.

Hence

(2(ha_1-h_1a))/(ab_1-a_1b)= (ab_1-a_1b)/(2(h_1b-hb_1))

color(magenta)(=>4(ha_1-h_1a)(h_1b-hb_1)=(ab_1-a_1b)^2)

This is the required condition.