Straight Lines
Straight Lines
Straight Lines
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2. SLOPE
The slope of a line is equal to the tangent of the angle which it makes with the positive side of x-axis and it is
generally denoted by m.
Thus if a line makes an angle with x-axis then its slope = m = tanθ
y y
o ) x
x
o
y 2 y1
The slope of a line joining two points (x1, y1) and (x2 , y2) is given by m = x x
2 1
a
The slope of a line ax + by + c = 0 is
b
Note :
(i) Slope of x axis or a line parallel to x axis is tan 0° = 0
(ii) Slope of y axis or a line parallel to y axis is tan 90° =
[1]
[2] Straight lines
b c
sinα = and p =
2 2
a +b a + b2
2
a 2x
a1x + b1 y + c1 a 2 x + b 2 y + c2 P
=± a 1x+ b1y+ c 1= 0
2 2 2 2
a +b
1 1 a +b
2 2
First write the equation of the lines so that the constant terms are positive. Then
(a) If a1a2 + b1b2 > 0 then on taking positive sign in the above bisectors equation we shall get the obtuse angle
bisector and on taking negative sign we shall get the acute angle bisector.
(b) If a1a2 + b1b2 < 0 then positive sign give the acute angle bisector and negative sign gives the obtuse angle
bisector.
(c) On taking positive sign we shall get equation of the bisector of the angle which contains the origin and
negative sign gives the equation of the bisector which does not contain origin.
Note : This is also the bisector of the angle in which origin lies ( since c1 , c2 are positive and it has been obtained
by taking positive sign)
16. FOOT OF THE PERPENDICULAR AND REFLECTION OF A POINT WITH RESPECT TO A LINE
(i) The foot (h, k) of the perpendicular drawn from the point (x1 , y1 ) on the line ax + by + c = 0 is given by
h x1 k y1 ax by c
1 2 12
a b a b
h x1 k y1 ax by c
(ii) The image (h, k) of the point (x1, y1) in the line ax + by + c = 0 is given by 2 1 2 12
a b a b
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