Point Slope Form Formula of a Line Last Updated : 10 Jan, 2024 Comments Improve Suggest changes Like Article Like Report In geometry, there are several forms to represent the equation of a straight line on the two-dimensional coordinate plane. There can be infinite lines with a given slope, but when we specify that the line passes through a given point then we get a unique straight line. Different forms of equations of straight line are: Slope-Intercept formPoint-Slope formStandard formIntercept formVertical formHorizontal form.So point-slope form is the form of the equation of the straight line which uses a point and the slope of the line to form the equation of the straight line. There exist many straight lines with the same slope but the straight line passing through that point is unique and the equation is also unique. In this article let us see how to form the equation of the straight line using the point-slope form in the 2D plane. Equation of a straight line if the point on the line is (a, b) and the slope of the line is m. y - b = m(x - a) Point-Slope formDerivation of the above formula: We know that slope is defined as slope = difference in y co-ords/ difference in x co-ords So, for the points P(x,y) and Q(a,b) the slope is m = (y - b)/(x - a) => m(x - a) = y - b => y - b = m(x - a) Where (x,y) is the points that satisfy the equation and lie on that line. It can be further extended to 2 point slope form if two points are given let two points be (x1, y1) and (x2, y2) then two point slope form is Slope is given as m = (y2 - y1)/(x2 - x1) Substituting m in one point slope form considering (x1,y1) as a point: y - y1 = (y2 - y1)/(x2 - x1) * (x - x1) Sample ProblemsQuestion1: Find the slope of the line that passes through (5,3), (6,4). Solution: The slope of the line when two points are given is m = y2 - y1/x2 - x1 = 4 - 3/6 - 5 = 1/1 =1 So slope of the line is 1 Question 2: Find the slope of the line when the equation of the line is 2x - 3y + 1. Solution: The slope of the line when equation of line is given in the form of ax + by + c is -a/b a = 2, b = -3 m = -(2/-3) So slope of the line is 2/3 Question 3: Find the equation of the line using point-slope form whose slope is 5 and point is (2,5) Solution: Given m = 5, (a,b) = (2,5) So using point-slope form : y-b = m(x-a) => y-5 = 5* (x-2) =>y -5 = 5x-10 =>5x-y-5 = 0 Equation of the line is 5x - y - 5 = 0 Question 4: Find the equation of the line using point-slope form whose slope is 2 and point is (1,3) Solution: Given m = 2 , (a,b) = (1,3) So using point-slope form : y-b = m(x-a) => y-3 = 2* (x-1) =>y -3 = 2x-2 =>2x-y+1 = 0 So the equation of the line is 2x - y + 1 = 0 Question 5: Find the equation of the line using point-slope form whose slope is 8 and point is (4,3) Solution: Given m = 8, (a,b) = (4,3) So using point-slope form : y-b = m(x-a) => y-3 = 8 (x-4) => y - 3 = 8x - 34 => 8x -y -31 = 0 Equation of the line is 8x - y -31 = 0 Question 6: Find the equation of the line passing through 2 points (3,6) and (4,8). Solution: Since 2 points are given 2 point slope form can be used: y-y1 = (y2-y1)/(x2-x1) * (x-x1) Given x1 = 3, y1 = 6, x2 = 4, y2 = 8 2 point slope form: y-y1 = (y2-y1)/(x2-x1) * (x-x1) => y -6 = (8-6 )/(4-3) * (x-3) => y-6 = 2*(x-3) => y -6 = 2x-6 => 2x-y =0 So equation of the line is 2x-y = 0 Question 7: Find the equation of the line passing through 2 points (1,5) and (4,17). Solution: Given x1 = 1, y1 = 5, x2 = 4, y2 = 17 2 point slope form : y-y1 = (y2-y1)/(x2-x1) * (x-x1) => y - 5 = (17-5)/(4-1) * (x-1) =>y-5 = 3 * (x-1) =>y -5 = 3x -3 => 3x - y + 2 = 0 So equation of the line is 3x - y + 2 = 0 Comment More infoAdvertise with us Next Article Writing Slope-Intercept Equations L lokeshpotta20 Follow Improve Article Tags : Mathematics School Learning Maths MAQ Maths-Formulas Similar Reads Geometry Geometry is a branch of mathematics that studies the properties, measurements, and relationships of points, lines, angles, surfaces, and solids. 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Cubes, sphere 3 min read LinesEquation of a Straight Line | Forms, Examples and Practice QuestionsThe equation of a line describes the relationship between the x-coordinates and y-coordinates of all points that lie on the line. It provides a way to mathematically represent that straight path.In general, the equation of a straight line can be written in several forms, depending on the information 10 min read Slope of a LineSlope of a Line is the measure of the steepness of a line, a surface, or a curve, whichever is the point of consideration. The slope of a Line is a fundamental concept in the stream of calculus or coordinate geometry, or we can say the slope of a line is fundamental to the complete mathematics subje 12 min read Angle between a Pair of LinesGiven two integers M1 and M2 representing the slope of two lines intersecting at a point, the task is to find the angle between these two lines. Examples: Input: M1 = 1.75, M2 = 0.27Output: 45.1455 degrees Input: M1 = 0.5, M2 = 1.75Output: 33.6901 degrees Approach: If ? is the angle between the two 4 min read Slope Intercept FormThe slope-intercept formula is one of the formulas used to find the equation of a line. The slope-intercept formula of a line with slope m and y-intercept b is, y = mx + b. Here (x, y) is any point on the line. It represents a straight line that cuts both axes. Slope intercept form of the equation i 9 min read Point Slope Form Formula of a LineIn geometry, there are several forms to represent the equation of a straight line on the two-dimensional coordinate plane. There can be infinite lines with a given slope, but when we specify that the line passes through a given point then we get a unique straight line. Different forms of equations o 6 min read Writing Slope-Intercept EquationsStraight-line equations, also known as "linear" equations, have simple variable expressions with no exponents and graph as straight lines. A straight-line equation is one that has only two variables: x and y, rather than variables like y2 or √x. Because it contains information about these two proper 10 min read Slope of perpendicular to lineYou are given the slope of one line (m1) and you have to find the slope of another line which is perpendicular to the given line. Examples: Input : 5 Output : Slope of perpendicular line is : -0.20 Input : 4 Output : Slope of perpendicular line is : -0.25 Suppose we are given two perpendicular line 3 min read What is the Point of Intersection of Two Lines Formula?If we consider two lines a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0, the point of intersection of these two lines is given by the formula:(x, y) = \left( \frac{b_1 c_2 \ - \ b_2 c_1}{a_1 b_2 \ - \ a_2 b_1}, \frac{c_1 a_2 \ - \ c_2 a_1}{a_1 b_2 \ - \ a_2 b_1} \right),The given illustration shows the i 5 min read Slope of the line parallel to the line with the given slopeGiven an integer m which is the slope of a line, the task is to find the slope of the line which is parallel to the given line. Examples: Input: m = 2 Output: 2 Input: m = -3 Output: -3 Approach: Let P and Q be two parallel lines with equations y = m1x + b1, and y = m2x + b2 respectively. Here m1 an 3 min read Minimum distance from a point to the line segment using VectorsGiven the coordinates of two endpoints A(x1, y1), B(x2, y2) of the line segment and coordinates of a point E(x, y); the task is to find the minimum distance from the point to line segment formed with the given coordinates.Note that both the ends of a line can go to infinity i.e. a line has no ending 10 min read Distance between two parallel linesGiven are two parallel straight lines with slope m, and different y-intercepts b1 & b2.The task is to find the distance between these two parallel lines.Examples: Input: m = 2, b1 = 4, b2 = 3 Output: 0.333333 Input: m = -4, b1 = 11, b2 = 23 Output: 0.8 Approach: Let PQ and RS be the parallel lin 4 min read Equation of a straight line passing through a point and making a given angle with a given lineGiven four integers a, b, c representing coefficients of a straight line with equation (ax + by + c = 0), the task is to find the equations of the two straight lines passing through a given point (x1, y1) and making an angle ? with the given straight line. Examples: Input: a = 2, b = 3, c = -7, x1 = 15+ min read Like