the equation ax²+2hxy+by²+2gx+2fx+c=0 represents a pair of straight lines intersecting on x axis
then what is the condition?

ans:- 2fgh=af²+ch²

please say me the way to get this answer
iam 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).