Example 21 - Find distance of 4x - y = 0 from P (4, 1) - Distance of a point from a line along a line

  1. Chapter 10 Class 11 Straight Lines
  2. Serial order wise
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Example 21 Find the distance of the line 4x – y = 0 from the point P (4, 1) measured along the line making an angle of 135° with the positive x-axis. There are two lines Line AB 4x – y = 0 Line CD making an angle 135° with positive x-axis Both lines meet at Q Point P(4, 1) is on line CD We need to find distance PQ. In PQ, P is (4,1) We need to find point Q Point Q is the intersection of line AB & CD Equation of AB is 4x – y = 0 Finding equation of line CD Slope of line CD = tan 135° = tan (180° − 45°) = − tan 45° = − 1 Also, Point P(4, 1) lies on the line CD Equation of a line passing through a point (x1, y1)& having slope m is (y − y1) = m(x − x1) Equation of line CD passes through point P(4,1) & having slope –1 is (y − 1) = − 1(x − 4) y − 1 = − x + 4 y + x = 4 + 1 x + y = 5 ∴ Equation of line CD is x + y = 5 Finding point Q Equation of AB : 4x − y = 0 Equation of CD : x + y = 5 Adding (1) & (2) 4x – y + x + y = 0 + 5 4x + x − y + y = 5 5x + 0 = 5 5x = 5 x = 5/5 = 1 Putting x = 1 in (1) 4x − y = 0 4(1) − y = 0 4 − y = 0 4 = y y = 4 Hence point Q (1, 4) Now we need to find distance between Q (1, 4) & P(4, 1) PQ = √((𝑥_2 − 𝑥_1 )^2 + (𝑦_1 − 𝑦_2 )^2 ) = √((1 − 4)^2 + (4 − 1)^2 ) = √(( − 3)^2 + (3)^2 ) = √(9 + 9) = √18 = √(9 ×2) = √(3^2×2) = = √(3^2 ) × √2 = 3√2 units Hence the required distance is 3√2 units

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