Ex 10.3, 16 - If p and q are lengths of perpendiculars - Distance of a point from a line

  1. Chapter 10 Class 11 Straight Lines
  2. Serial order wise
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Ex 10.3, 16 If p and q are the lengths of perpendiculars from the origin to the lines x cos θ – y sin θ = k cos 2θ and x sec θ + y cosec θ = k, respectively, prove that p2 + 4q2 = k2 The given lines are x cos θ – y sin θ = k cos 2θ x sec θ + y cosec θ = k cos 2θ The distance (d) of a line Ax + By + C = 0 from a point (x1, y1) is given by d = |𝐴𝑥_1 + 𝐵𝑦_1 + 𝐶|/√(𝐴^2 + 𝐵^2 ) Taking (1) x cos θ – y sin θ = k cos 2θ x cos θ – y sin θ – k cos 2θ = 0 The above equation is of the form Ax + By + C = 0 Where A = cos θ , B = – sin θ , C = – k cos 2θ Also, p is the perpendicular distance from origin to line (1) Here x1 = 0 , y1 = 0 , & d = p Putting values p = |cos⁡〖θ(0) + ( − sin⁡θ )(0) − k cos⁡2θ 〗 |/√((cos⁡θ )^2 + ( − sin⁡θ )2) p = |0 + 0 − k cos⁡2θ |/√(cos^2⁡〖θ + sin^2 θ〗 ) p = (𝑘 cos⁡2θ)/√(cos^2⁡〖θ + sin^2 θ〗 ) p = (𝑘 cos⁡2θ)/√1 p = (k cos⁡2θ)/1 p = k cos 2θ From line (2) x sec θ + y cosec θ = k x (1/cos⁡θ ) + y (1/sin⁡θ ) = k (x sin⁡〖θ + y cos⁡θ 〗)/(cos⁡θ sin⁡θ ) = k x sin θ + y cos θ = k cos θ sin θ sin θ x + cos θ y – k cos θ sin θ = 0 The above equation is of the form Ax + By + C = 0 where A = sin θ , B = cos θ , C = – k cos θ sin θ The distance (d) of a line Ax + By + C = 0 from a point (x1, y1) is given by d = |𝐴𝑥_1 + 𝐵𝑦_1 + 𝐶|/√(𝐴^2 + 𝐵^2 ) Given that q is the perpendicular distance from origin to line (2) Here x1 = 0 , y1 = 0 , & d = q Putting values q = |sin⁡〖θ(0) + 〖 cos〗⁡〖θ(0) − k sin⁡〖θ cos⁡θ 〗 〗 〗 |/√(sin^2 θ + cos^2 θ ) q = |0 + 0 − 𝑘 sin⁡〖𝜃 cos⁡𝜃 〗 |/√1 q = | − 𝑘 sin⁡〖𝜃 cos⁡𝜃 〗 |/1 q = k sin θ cos θ We need to show p2 + 4q2 = k2 Taking LHS p2 + 4q2 Putting values from (3) & (4) = (k cos 2θ)2 + (2k sin θ cos θ)2 = k2 (cos 2θ)2 + k2(2sin θ cos θ)2 = k2 cos2 2θ + k2 (sin 2θ)2 = k2 cos2 2θ + k2 sin2 2θ = k2 (cos2 2θ + sin2 2θ) = k2 (1) = k2 = R.H.S Hence L.H.S = R.H.S Hence proved

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Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 8 years. He provides courses for Maths and Science at Teachoo. You can check his NCERT Solutions from Class 6 to 12.