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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
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,
Ashwin (IIT Madras).
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