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Question

Show that a homogeneous equation of degree two in x and y, i.e., ax2+2hxy+hy2=0 represents a pair of lines passing through the origin if h2ab0.

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Solution

Consider a homogeneous equation of degree two in x and y, ax2+2hxy+hy2=0...(1)

In this equation at least one of the coefficients a,borh is non zero.

We consider two cases.

Case I : If b=0, then the equation of lines are x=0 and (ax+2hy)=0.
These lines passes through the origin.

Case II : b0, Multiplying both the sides of equation (1) by b, we get
abx2+2hxyb+hby2=0

hby2+2hbxy=abx2

To make L.H.S a complete square, we add h2x2 on both the sides.

hby2+2hbxy+h2x2=abx2+h2x2

(by+hx)2=(h2ab)x2

(by+hx)2=[(h2ab)x]2

(by+hx)2[(h2ab)x]2=0

[(by+hx)(h2ab)x][(by+hx)+(h2ab)x]=0

It is the joint equation of two lines (by+hx)(h2ab)x=0 and (by+hx)+(h2ab)x=0

That is, (h+h2ab)x+by=0 and (hh2ab)x+by=0

These lines passes through the origin.

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