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Transcript of IB-05pair-straightlines(17-24)
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1
5. PAIR OF STRAIGHT LIN ES
Synopsis:
1. If h2
ab, then ax2
+ 2hxy + by2
= 0 represent a pair of straight lines passing through the origin.2. If h
2 < ab, then ax2 + 2hxy + by2 = 0 represents two imaginary lines having real point of
intersection, the origin.
3. If h2= ab, then ax
2+ 2hxy + by
2= 0 represents coincident lines.
4. If h2
> ab, then ax2
+ 2hxy + by2
= 0 represents two real and different lines.
5. If ax2
+ 2hxy + by2
= 0 represents a pair of lines,
i) sum of slopes of lines is2h/b.
ii) the product of slopes of lines is a/b
6. If is an angle between the lines represented by ax2 + 2hxy + by2 = 0, then
i) cos =22 h4)ba(
ba
++
ii) sin =22
2
h4)ba(
abh2
+
iii) tan =ba
abh2 2
+
7. If is the acute angle between the lines represented by ax2 + 2hxy + by2 = 0, then
i) cos =22
h4)ba(
ba
+
+
ii) tan =ba
abh2 2
+
8. If = 0, then the two lines will be coincident h2 = ab.
9. If = /2, then the two lines will be perpendiculara + b = 0 coefficient of x2 + coefficient of y2 = 0.
10. The equation to the pair of lines passing through (, ) and parallel to ax2 + 2hxy + by2 = 0 isa(x)2 + 2h(x )(y ) + b(y )2 = 0
11. The equation to the pair of lines passing through the origin and perpendicular to ax2
+ 2hxy + by2
= 0 is
bx22hxy + ay2 = 0.
12. The equation to the pair of lines passing through the point (, ) and perpendicular to ax2 + 2hxy + by2 = 0 isb(x )2 2h(x )(y ) + a(y )2 =0.
13. The equation of bisectors of angles between the lines a1x + b1y + c1 = 0. a2x + b2y + c2 = 0 are
122
1
111
ba
cybxa
+
++
222
2
222
ba
cybxa
+
++= 0 If c1c2>0 then the line bisecting the angle containing the origin between
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Pair of straight lines
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the lines is
122
1
111
ba
cybxa
+
++
222
2
222
ba
cybxa
+
++= 0 and the line bisecting the other angle is
122
1
111
ba
cybxa
+
+++
2222
222
ba
cybxa
+
++= 0
14. The equation to the pair of bisectors of angles between the pair of lines ax2
+ 2hxy + by2
= 0 is h(x2 y
2) (a
b)xy = 0.
15. Every pair of lines are equally inclined to either of its angle bisectors.
16. Two pairs of concurrent lines are equally inclined to each other if both the pairs of lines have the same angle
bisectors.
17. The product of the perpendiculars from (,) to the pair of lines ax2 + 2hxy + by2 = 0 is22
22
h4)ba(
bh2a
+
++.
18. The area of the triangle formed by the lines ax2
+ 2hxy + by2
= 0 and the line lx + my + n = 0 is
22
22
blhlm2am
abhn
+
.
19. The line ax + by + c = 0 and the pair of lines (ax + by)2 3(bx ay)
2= 0 form an equilateral triangle and its
area is)ba(3
c22
2
+sq. unit.
20. The equation to the pair of lines passing through the origin and forming an equilateral triangle with
the line ax + by + c = 0 is (ax + by)2
3(bx ay)2
= 0.
21. If ax2
+ 2hxy + by2
+ 2gx + 2fy + c = 0 represents a pair of lines then
i) abc + 2fghaf2bg2ch2 = 0ii) h
2 ab, g2 ac, f2bc.
iii) The point of intersection of the lines is
22 hab
afgh,
hab
bghfwhere h
2 ab.
22. The difference of the slopes of the lines represented by ax2
+ 2hxy + by2
= 0 is .b
abh2 2
23. The condition that the slope of the lines represented by ax2
+ 2hxy + by2
= 0 is the ratio p:q is ab(p + q)2
=
4h2pq.
24. If a1a2 + b1b2 > 0, then the acute angle bisector of the angles between a1x + b1y + c1 = 0, a2x + b2y +
c2 = 0 is2
22
2
222
12
12
111
ba
cybxa
ba
cybxa
+
++++
++ = 0.
25. The equation to the pair of lines passing through the origin and each is at a distance of d from
(x1, y1) is (xy1 x1y)2
= d2(x
2+ y
2).
26. The condition that the lines ax2
+ 2hxy + by2
= 0, lx + my + n = 0 to form an isosceles triangle of
vertex origin is h(l2 m
2) = (a b)lm.
27. If ax2
+ 2hxy + by2
= 0 are two sides of a parallelogram and lx + my + n = 0 is one diagonal, then
the equation to the other diagonal is y(bl hm) = x(am hl).
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Pair of straight lines
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28. The length of the intercept on x axis cut by the pair of the lines ax2
+ 2hxy + by2
+ 2gx + 2fy+c =
0 is .a
acg2 2
29. The length of the intercept on y axis cut by the pair of lines ax2 + 2hxy + by2 + 2gx + 2fy + c = 0
isb
bcf2 2 .
30. The condition that the pair of lines represented by ax2
+ 2hxy + by2
+ 2gx + 2fy + c = 0 intersect on
x axis is g2
= ac, on y axis is f2
= bc.
31. The condition that the pair of lines represented by ax2
+ 2hxy + by2
+ 2gx + 2fy + c = 0 is
22
22
h4)ba(
cf2g2bh2a
+
+++++
32. The lines represented by ax2
+ 2hxy + by2
= 0,ax2
+ 2hxy + by2
+ 2gx + 2fy + c = 0 to form a
i) parallelogram if a + b0, (a b)fg + h + h(f2 g2)0
ii) rectangle if a + b = 0, (a b)fg + h(f2 g2)0
iii)rhombus if a + b0, (a b)fg + h(f2 g2) = 0iv)square of a + b = 0, (a b)fg + h(f2 g2) = 0
35.The area of parallelogram formed by ax2 + 2hxy + by2 = 0,ax2 + 2hxy + by2 + 2gx + 2fy + c = 0 is
abh2
c
2
36.The condition that one of the lines represented by S = 0 bisects the angle between the co ordinateaxes is (a + b)2 = 4h2.
37.Equation to the pair of angular bisectors of the lines represented by SI = 0 isba
)yy()xx( 212
1
=
h
)yy)(xx( 11 where (x1, y1) is point of intersection of lines of SI= 0.
38.I) If is the angle between the lines joining the origin to the points of intersection of the circle x2
+ y2
= a2
and the line lx + my + n = 0. Then Cos22 mla
n
2 +=
.
ii) These lines are perpendicular then a2(l
2+ m
2) = 2n
2.
37. The line joining the origin to the points of intersection of the line kx + hy = 2hk with the curve
(xh)2
+ (yk)2
= c2
are at right angles if h2
+k2
= c2
.38. Point of intersection of diagonals of rectangle formed by the pairs a1x
2+ b1x + c1 = 0 and a2y
2+
b2y + c2 = 0 is
2
2
1
1
a2
b,
a2
b.
39. If equation ax2
+ 2hxy + by2
= 0 represents two sides of a triangle whose orthocentre is (c, d) then
equation of third side is (a + b)(cx + dy) = bc2 2hcd + ad2.
40. The lines joining the origin to the points of intersection of the curves a1x2
+ 2h1xy + b1y2
+ 2g1x =
0 and a2x2
+ 2h2xy + b2y2
+ 2g2x = 0 will be at right angles if2
1
22
11
g
g
ba
ba=
++
.
