Straight line and Standard Form of Straight Lines
Different Standard Forms .of the Equation of a Straight Line:
1. General form:
Ax + By + C = 0
where A, B, C are any real numbers not all zero.
2. Equation of x-axis is y = O.
3. Equation of y-axis is x = O.
4. Equation of a line parallel to x-axis and at a distance 'b' from it is y = b.
5. Equation of a line parallel to y-axls and at a distance 'a' from it is x = a.
6. Equation of a line parallel to x-axis and passing through (a, b) is y = b.
7. Equation of a line perpendicular to x-axis and passing through (a, b) is x = a.
8. Equation of a line parallel to y-axis and passing through (a, b) is x= a.
9. Equation of a line perpendicular to y-axis and passing through (a, b) is y = b.
10. (Gradient tangent form or slope Intercept form)
y = mx + c
It is the equation of a straight line which it cuts off an intercept c on y-axis and makes and angle with the positive direction (anticlockwise) of x-axis such that tan 8 = m. The number m is called slope or (the gradient) of this line. If the straight line passes through origin, then c = 0 and equation becomes v = mx.
11. Intercept form: Equation of a straight line which cuts off given intercepts a and b from the axis
is the required a b equation of line.
Let a straight line AB meet the x-axis and y-axis in A and B respectively. Then
(i) Length of directed segment OA is called the intercept cut by the line on x-axis or x-intercept.
(ii) Length of directed segment OB is called the intercept cut by the line on y-axis or y-intercept.
(iii) AB is called the portion of line intercepted between the axes.
(iv) If OA = a, then A is (a, 0); if OB = b, then B is (0, b).
Rule for signs of intercept:
(i) The intercept on the x-axis is positive if measured to the right of the origin, and negative if measured to the left.
(ii) The intercept on the y-axis is positive if it is measured above the origin, and negative if measured below.
Cor. I. If the line is parallel to the x-axis, then a → ∞, x/a → 0 and the equation becomes 0 + y/b = 1 or y = b.
Cor. II. If the line is parallel to the y-axis, then b → ∞ y/b→0, and the equation becomes x/a+0=1 or x = a.
12. y = mx is the equation of any straight line passing through the origin.
13. Equation of line passing through origin and makes equal angle with both the axis is y = ±x.
14. Intercept form:
If the equation of straight line which it cut off intercepts a and b on the axis of x and y respectively.
15. Normal from (perpendicular from):
x cos a + y sin a = p
It is the equation of a straight line on which the length of the perpendicular from the origin is p and a is the angle which, this perpendicular makes with the positive direction of x-axis.
16. One point from: y - y1 = m (x - x1)
It is the equation of a straight line passing through a given point (x1 y1) and having slope m.
17. Parametric equation:
It is the equation of a straight line passes through a given point A(x1, y1) and makes an angle e with x-axis.
The co-ordinates (x, y) of any point P on this line are (x1 + r cos q, y1 + r sin q). The distance of this point P from the given point A is
18. Two points from:
It is the equation of a straight line passing through two given points (x1, y1) and (x2. y2), where (y2-y1)/(x2-x1) is its slope.
19. The equations of the lines through (x1, y1) and making an angle, with the line ax + by + c = 0 are y - y1 = m1 (x - x1) where m1,
and y - y1 = m2 (x - x1) where m2
where m = tan q = - is the slope of the given line.
Note that m1 = tan (q - f) and m2 = tan (q + f).
Slope of line Ax + By + C = 0 ............... (i)
is given by m =- A/B = and intercepts on X and y-axis are -C/A and
Another method: The line (i) meets x-axis at a point for which Y = 0.
\ Putting y = 0 in (i). we get x = - C/A = intercept on x-axis.
Also the line (i) meets y-axis at a point for which x = 0.
\ Putting x = 0 in (i) we get Y = - C/B = intercept on y-axis.
Point of intersection of two lines a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0 is given by
Collinearity of any Three Points A, B, C:
(1) Find the equation of straight line joining any two points and show that the coordinates of the third point satisfy it.
(2) Find the slope of lines AB and AC and show that slope of AB = slope of AC.
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