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#?

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Answer 1 · 2018-08-06T13:44:12

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.