show that the homogeneous equation of degree two always represents a pair of straight lines passing through origin also find the angle between them
Answers
Answer:
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Answer:
ANSWER
Consider a homogeneous equation of degree two in x and y, ax
2
+2hxy+hy
2
=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 : b
=0, Multiplying both the sides of equation (1) by b, we get
abx
2
+2hxyb+hby
2
=0
⇒hby
2
+2hbxy=−abx
2
To make L.H.S a complete square, we add h
2
x
2
on both the sides.
⇒hby
2
+2hbxy+h
2
x
2
=−abx
2
+h
2
x
2
⇒(by+hx)
2
=(h
2
−ab)x
2
⇒(by+hx)
2
=[(
h
2
−ab
)x]
2
⇒(by+hx)
2
−[(
h
2
−ab
)x]
2
=0
⇒[(by+hx)−(
h
2
−ab
)x]⋅[(by+hx)+(
h
2
−ab
)x]=0
It is the joint equation of two lines (by+hx)−(
h
2
−ab
)x=0 and (by+hx)+(
h
2
−ab
)x=0
That is, (h+
h
2
−ab
)x+by=0 and (h−
h
2
−ab
)x+by=0
.................These lines passes through the origin.
These lines passes through the origin. support me............