![]() If we multiply the above two factors together, we get a more general equation to a pair of straight lines has the formĪx 2 + 2 hxy + by 2 + 2 gx + 2 f y + c = 0. ![]() General form of Pair of Straight LinesĬonsider the equations of two arbitrary lines l 1 x + m 1 y + n 1 = 0 and l 2 x + m 2 y + n 2 = 0 The combined equation of the two lines is The perpendiculars from P ( p, q ) to the line y − m 1 x = 0 is equal to the perpendicular from P ( p, q) to y − m 2 x = 0Ĥ. Let P ( p, q ) be any point on the locus of bisectors. We know that the equation of bisectors is the locus of points from which the perpendicular drawn to the two straight lines are equal. Equation of the bisectors of the angle between the linesĪx 2 + 2 hxy + by 2 = 0 Let the equations of the two straight lines be y − m 1 x = 0 and y − m 2 x = 0 ![]() The lines are not real (imaginary), if m 1 and m 2 are not real, that is if h 2 − ab The lines are real and coincident, if m 1 and m 2 are real and equal, that is if h 2 − ab = 0.The lines are real and distinct, if m 1 and m 2 are real and distinct, that is if h 2 − ab > 0.The above equation suggests that the general equation of a pair of straight lines passing through the origin with slopes m 1 and m 2, ax 2 + 2 hxy + by 2 = 0 is a homogenous equation of degree two, implying that the degree of each term is 2.Īs a consequence of this formula, we can conclude that Both the lines in this pair pass through the origin. L 2 = 0 represents the pair of straight lines L 1 = 0 and L 2 = 0.If P ( x 1, y 1) lies either on L 1 = 0 or L 2 = 0, then P ( x 1, y 1) satisfies the equation ( L 1 ) ( L 2) = 0, and no other point satisfies L 1 Similarly, if P ( x 1, y 1) is on L 2 then L 2 = 0. If P ( x 1, y 1) is a point on L 1, then it satisfies the equaiton L 1 = 0. Let L 1 ≡ a 1 x + b 1 y + c 1 = 0 and L 2 ≡ a 2 x + b 2 y + c 2 = 0, be separate equations of two straight lines. Hence we study pair of straight lines as a quadratic equations in x and y. As we see that a linear equation in x and y represents a straight line, the product of two linear equations represent two straight lines, that is a pair of straight lines. This means when two lines are perpendicular to each other, the product of their slopes is -1, i.e., if m is the slope of L 1, then the slope L 2 ⏊ to it is (-1/m).The equations of two or more lines can be expressed together by an equation of degree higher than one. Let’s say this point M divides PQ in the ratio of n:1. Let M be the point on the line segment joining P and Q. Let points P (x 1, y 1 ) and Q (x 2, y 2 ) be any two points on the curve represented by ax + by + c = 0. Let ax + by + c = 0 be a first-degree equation in x,y where a, b, and c are constant. Let’s throw light on the coordinate geometry to prove that every first-degree equation in x, y represents a straight line. We define a straight line as a curve where every point on the line segment joining any two points lies on it. Oblique or slanting lines: The lines drawn in a slanting position are termed as oblique or slanting lines. Vertical lines: The lines drawn vertically Horizontal lines: The lines drawn horizontally A straight line can be further differentiated into horizontal, vertical, or slanted. A straight line is devoid of curves and does not have any curve on it. It can also be said to be a combination of endless points joined on both sides of a point. The straight line extends to both sides.Ī straight line can be defined as an endless one-dimensional figure with no width. Now if we add those sides then we would be looking at full geometric relations where adding up lines will make up different shapes, that is a scientific notion but Mathematically any fixed line or curve on the line segment can be defined as a straight line.Ī line with zero or no curves or a structure with infinite length devoid of any curves is defined as a straight line. Now several examples of straight lines would be railway tracks, ruler (to measure), even parts of a mirror can be called a straight line. We come across many things daily which are completely straight and aligned along the similar lines or in parallel setting with the other line of the same length or different measurements but straight line.
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