Show that equation 2x2 - xy - 3y2 - 6x + 19 - 20 = 0 represents a pair of lines.
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Asked on December 27, 2019 by
Ivy Dontask
Show that the equation 2x
2
−xy−3y
2
−6x+19y−20=0 represents a pair of intersecting lines. Show further that the angle between them is tan
−1
5
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ANSWER
Given equation 2x
2
−xy−3y
2
−6x+19y−20=0
ax
2
+2hxy+by
2
+2gx+2fy+c=0
a=2,h=
2
−1
,b=−3,f=
2
19
,c=−20
abc+2fgh−af
2
−bg
2
−ch
2
=0 condition for intersection
=2(−3)(−20)+2(
2
19
)(−3)(
2
−1
)−2(
2
19
)
2
−(−3)(−3)
2
−(−20)(
2
−1
)
2
=120+
2
57
−2(
4
361
)+27+20(
4
1
)
=120+
2
57
−(
2
361
)+27+5
=
2
(240+57−361+54+10)
=0
hence the lines are intersecting
let θ angle between them:
θ=tan
−1
∣
∣
∣
∣
∣
∣
2
(a+b)
(h
2
−ab)
∣
∣
∣
∣
∣
∣
θ=tan
−1
∣
∣
∣
∣
∣
∣
∣
2
((−
2
1
)
2
−2(−3)/(2+(−3))
∣
∣
∣
∣
∣
∣
∣
=tan
−1
∣
∣
∣
∣
∣
∣
2
((
4
1
)+6)
/(−1)
∣
∣
∣
∣
∣
∣
=tan
−1
∣
∣
∣
∣
∣
∣
2
(
4
25
)
∣
∣
∣
∣
∣
∣
=tan
−1
∣
∣
∣
∣
∣
2
2(5)
∣
∣
∣
∣
∣
=tan
−1
5
Hence proved .
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