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# the equation "ax2+2hxy+by2+2gx+2fx+c=0" represents a pair of straight lines intersecting on x axisthen what is the condition?ans:- 2fgh=af2+ch2 please say me the way to get this answeriam thankful for those who give me answer

290 Points
10 years ago

Hi Nikhil,

Let the pair of straight lines intersect at a point (p,0) on the x-axis. Hence this point must satisfy the gievn eqn.

Note that in your question, it should have been 2fy instead of 2fx.

So, ap2+0+0+2gp+0+c = 0.

Hence ap2+2gp+c = 0 is the QE, which should have co-incident roots.

Hence g2 = ac ----------(from D = 0).

Now for the given equation to represent a pair of St lines, the determinant

a   h   g

h   b   f

g   f   c

must be zero.

And hence a(bc-f2) - h(hc-fg) + g(hf-bg) = 0

ie abc - af2 - ch2 + 2fgh - bg2 = 0. ---------- (but g2=ac)

So 2fgh = af2+ch2 ------- (the necessary condition).

Hope this helps.

Wish you all the best.

Regards,