Find the value of ' p ' if the lines, 5 x - 3 y + 2 = 0 and 6 x - p y + 7 = 0 are perpendicular to each other. Hence find the equation of a line passing through - 2 , - 1 and parallel to 6 x - p y + 7 = 0 .

We know that the slope of a line of the form a x + b y + c = 0 , is = - a b The slope of the line 5 x - 3 y + 2 = 0 , is = - 5 - 3 = 5 3 The slope of the line 6 x - p y + 7 = 0 , is = - 6 - p = 6 p We know that when two lines are perpendicular multiplication of slopes of those two lines is equal to - 1 So, 5 3 × 6 p = - 1 5 3 × 6 = - p 10 = - p p = - 10 Then equation becomes 6 x + 10 y + 7 = 0 The slope of the line 6 x + 10 y + 7 = 0 is = - 6 10 = - 3 5 Equation of the line with the slope m and passing through the point x 1 , y 1 is y - y 1 = m x - x 1 The equation of the line with the slope - 3 5 and passing through the point - 2 , - 1 is given below. y - - 1 = - 3 5 x - - 2 y + 1 = - 3 5 x + 2 5 y + 1 = - 3 x + 2 5 y + 5 = - 3 x - 6 3 x + 5 y + 5 + 6 = 0 3 x + 5 y + 11 = 0 Hence, the required equation is 3 x + 5 y + 11 = 0 .

E M B I B E

Mathematics Crash Course (Based on Revised Syllabus-2023)>Chapter -1 - Coordinate Geometry>Exercise>Q 5

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