Show that every homogeneous equation of degree two in x and y , i.e., ax² +2hxy + by² = 0 represents a pair of lines passing through origin if h²- ab ≥ 0.
Answers
Answer:
Let the two lines be y = mı,x and y through origin.
y - mx= 0 and y – m2x = 0
= (y – mịx)(y – m,x)
= 0
y – (mı + m2)xy + m, m2x? = 0; Where m,and m, are slopes of lines.
Then
y – (sum of slopes)xy +(product of slopes)x2 = 0
It represents pair of straight lines passing through origin.
Now,ax + 2hxy + by? = 0
2h
→ y? + b ху + a x2 = 0
b
So
mi + m2 = and m m2 = a b
A.M is greater than G.M
So,
mi + m2
2 > m; m2 = (m + m2)² > 4m, m,
AI 4a b 4h2 b2
h> ab
h - ab >0
-2h
b=m2x
Answer:
Answer
Let two lines y=m
1
,x and y=m
2
x passess through origin.
⇒y−m
1
x=0 and y−m
2
x=0
⇒(y−m
1
x)(y−m
2
x)=0
∴y
2
−(m
1
+m
2
)xy+m
1
m
2
x
2
=0; Where m
1
and m
2
are slopes of lines.
∴y
2
−(sum of slopes)xy+(product of slopes)x
2
=0
It represents pair of straight lines passing through origin.
Now,
ax
2
+2hxy+by
2
=0
⇒y
2
+
b
2h
xy+
b
a
x
2
=0
⇒m
1
+m
2
=
b
−2h
and m
1
m
2
=
b
a
⇒A.M≥G.M
⇒
2
m
1
+m
2
≥
m
1
m
2
⇒(m
1
+m
2
)
2
≥4m
1
m
2
⇒
b
2
4h
2
≥
b
4a
⇒h
2
≥ab
∴h
2
−ab≥0