Notes On Equations of Lines (Part I) - CBSE Class 11 Maths
If a point lies on a line, then it must satisfy the equation of the line. Consider a horizontal line L that intersects the positive side of the Y-axis at a distance 'a' from the X-axis. The y coordinate of every point on line L at any position is 'a'. Equation of line l: y = a Consider another horizontal line S that intersects the negative side of the Y-axis at a distance 'a' from the X-axis. The y coordinate of every point on line S at any position is minus 'a'. Equation of line s: y = -a Consider a vertical line L that intersects the positive side of the X-axis at a distance 'b' from the Y-axis. The x coordinate of every point on line L at any position is 'b'. Equation of line l: x = b Consider another vertical line S that intersects the negative side of the X-axis at a distance 'c' from the Y-axis. The x coordinate of every point on line S at any position is minus 'c'. Equation of line s: y = -c Point-Slope Form Consider a non-vertical line L passing through a given point P(x1, y1). Let m be the slope of line L. Let A(x, y) be any arbitrary point on line L. The slope of a line passing through (x1, y1) and (x2, y2) is (y2 - y1)/(x2 - x1). Here, slope (m) = (y - y1)/(x - x1) ⇒ y - y1 = m (x - x1) ….(1) Equation 1 represents the point-slope form of the equation of a line passing through a given point (x1, y1) and with slope m. Two-Point Form Consider a non-vertical line L passing through two given points P and Q. Let A be any arbitrary point on line L. Three given points A, B and C are collinear if: Slope of AB = Slope of BC Thus, A, P and Q are collinear if: Slope of PA = Slope of PQ  Slope of PA = (y - y1)/ (x - x1) Slope of PQ = (y2 - y1)/ (x2 - x1) ⇒ (y - y1)/ (x - x1) = (y2 - y1)/ (x2 - x1) This equation represents the two-point form of the equation of a line passing through the given points, P and Q.

#### Summary

If a point lies on a line, then it must satisfy the equation of the line. Consider a horizontal line L that intersects the positive side of the Y-axis at a distance 'a' from the X-axis. The y coordinate of every point on line L at any position is 'a'. Equation of line l: y = a Consider another horizontal line S that intersects the negative side of the Y-axis at a distance 'a' from the X-axis. The y coordinate of every point on line S at any position is minus 'a'. Equation of line s: y = -a Consider a vertical line L that intersects the positive side of the X-axis at a distance 'b' from the Y-axis. The x coordinate of every point on line L at any position is 'b'. Equation of line l: x = b Consider another vertical line S that intersects the negative side of the X-axis at a distance 'c' from the Y-axis. The x coordinate of every point on line S at any position is minus 'c'. Equation of line s: y = -c Point-Slope Form Consider a non-vertical line L passing through a given point P(x1, y1). Let m be the slope of line L. Let A(x, y) be any arbitrary point on line L. The slope of a line passing through (x1, y1) and (x2, y2) is (y2 - y1)/(x2 - x1). Here, slope (m) = (y - y1)/(x - x1) ⇒ y - y1 = m (x - x1) ….(1) Equation 1 represents the point-slope form of the equation of a line passing through a given point (x1, y1) and with slope m. Two-Point Form Consider a non-vertical line L passing through two given points P and Q. Let A be any arbitrary point on line L. Three given points A, B and C are collinear if: Slope of AB = Slope of BC Thus, A, P and Q are collinear if: Slope of PA = Slope of PQ  Slope of PA = (y - y1)/ (x - x1) Slope of PQ = (y2 - y1)/ (x2 - x1) ⇒ (y - y1)/ (x - x1) = (y2 - y1)/ (x2 - x1) This equation represents the two-point form of the equation of a line passing through the given points, P and Q.

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