**Co-Ordinate Straight Line Geometry**

A line is simply an object in geometry that is characterized as a straight, thin, one-dimensional, zero width object that extends on both sides to infinity. A straight line is essentially just a line with no curves. Most of the time, when we speak about lines, we are talking about straight lines! Here are some examples of straight lines

The intercepts of a line are the points where the line intercepts, or crosses, the horizontal and vertical axes. A straight line moves so that the sum of the reciprocals of its intercepts made on axes is constant.

(i) The slope or gradient of a straight line is the trigonometric tangent of the angle θ which the line makes with the positive directive of x-axis.

(ii) The slope of x-axis or of a line parallel to x-axis is zero.

(iii) The slope of y-axis or of a line parallel to y-axis is undefined.

(iv) The slope of the line joining the points (x_{1}, y_{1}) and (x_{2}, y_{2}) is

m = (y_{2} – y_{1})/(x_{2} – x_{1}).

(v) The equation of x-axis is y = 0 and the equation of a line parallel to x-axis is y = b.

(vi) The equation of y-axis is x = 0 and the equation of a line parallel to y-axis is x = a.

(vii) The equation of a straight line in

- slope-intercept form: y = mx + c where m is the slope of the line and c is its y-intercept;
- point-slope form: y – y1 = m (x – x
_{1}) where m is the slope of the line and (x_{1}, y_{1}) is a given point on the line; - symmetrical form: (x – x
_{1})/cos θ = (y – y_{1})/sin θ = r, where θ is the inclination of the line, (x_{1}, y) is a given point on the line and r is the distance between the points (x, y) and (x_{1}, y_{1}); - two-point form: (x – x
_{1})/(x_{2}– x_{1}) = (y – y_{1})/(y_{2}– y_{1}) where (x_{1}, y_{1}) and (x_{2}, y_{2}) are two given points on the line; - intercept form: x/a + y/b = 1 where a = x-intercept and b = y-intercept of the line;
- normal form: x cos α + y sin α = p where p is the perpendicular distance of the line from the origin and α is the angle which the perpendicular line makes with the positive direction of the x-axis.
- general form: ax + by + c = 0 where a, b, c are constants and a, b are not both zero.

(viii) The equation of any straight line through the intersection of the lines a_{1}x + b_{1}y + c_{1} = 0 and a_{2}x + b_{2}y + c_{2} = 0 is a_{1}x + b_{1}y + c + k(a_{2}x + b_{2}y + c_{2}) = 0 (k ≠ 0).

(ix) If p ≠ 0, q ≠ 0, r ≠ 0 are constants then the lines a_{1}x + b_{1}y + c_{1} = 0, a_{2}x + b_{2}y + c_{2} = 0 and a_{3}x + b_{3}y + c_{3} = 0 are concurrent if P (a_{1}x + b_{1}y + c_{1}) + q (a_{2}x + b_{2}y + c_{2}) + r (a_{3}x + b_{3}y + c_{3}) = 0.

(x) If θ be the angle between the lines y= m_{1}x + c and y = m_{2}x + c_{2} then tan θ = ± (m_{1} – m_{2})/(1 + m_{1} m_{2});

(xi) The lines y= m_{1}x + c_{1} and y = m_{2}x + c_{2} are

(a) parallel to each other when m_{1} = m_{2};

(b) perpendicular to one another when m_{1} ∙ m_{2} = – 1.

(xii) The equation of any straight line which is

(a) parallel to the line ax + by + c = 0 is ax + by = k where k is an arbitrary constant;

(b) perpendicular to the line ax + by + c = 0 is bx – ay = k_{1} where k_{1} is an arbitrary constant.

(xiii) The straight lines a_{1}x + b_{1}y + c_{1} = 0 and a_{2}x + b_{2}y + c_{2} = 0 are identical if a_{1}/a_{2} = b_{1}/b_{2} = c_{1}/c_{2}.

(xiv) The points (x_{1}, y_{1}) and (x_{2}, y_{2}) lie on the same or opposite sides of the line ax + by + c = 0 according as (ax_{1} + by + c) and (ax_{2} + by_{2} + c) are of same sign or opposite signs.

(xv) Length of the perpendicular from the point (x_{1}, y_{1}) upon the line ax + by + c = 0 is|(ax_{1} + by_{1} + c)|/√(a^{2} + b^{2}).

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