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Misc 23 - Prove that product of 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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Misc 23 Prove that the product of the lengths of the perpendiculars drawn from the points (root a^2-b^2,0) and (-root a^2-b^2,0) to the line x/a cos theta +y/b sin theta =1 = b2 Let p1 be the perpendicular distance from point A(√(π‘Ž^2 βˆ’ 𝑏^2 ), 0) to the line π‘₯/π‘Žcos ΞΈ + 𝑦/π‘Ž sin ΞΈ = 1 & p2 be the perpendicular distance from point B( βˆ’ √(π‘Ž^2 βˆ’ 𝑏^2 ), 0) to the line π‘₯/π‘Žcos ΞΈ + 𝑦/π‘Žsin ΞΈ = 1 We need to show p1 Γ— p2 = b2 Calculating p1 & p2 Given line is π‘₯/π‘Žcos ΞΈ + 𝑦/𝑏sin ΞΈ = 1 (cosβ‘πœƒ/π‘Ž)x + (sinβ‘πœƒ/𝑏)y βˆ’ 1 = 0 = (π‘Ž^2 〖𝑠𝑖𝑛〗^2 πœƒ + 𝑏^2 γ€–π‘π‘œπ‘ γ€—^2 πœƒ)/(𝑏^2 γ€–π‘π‘œπ‘ γ€—^2 πœƒ + π‘Ž^2 〖𝑠𝑖𝑛〗^2 πœƒ) Γ— 𝑏^2 = (𝑏^2 γ€–π‘π‘œπ‘ γ€—^2 πœƒ + π‘Ž^2 〖𝑠𝑖𝑛〗^2 πœƒ)/(𝑏^2 γ€–π‘π‘œπ‘ γ€—^2 πœƒ + π‘Ž^2 〖𝑠𝑖𝑛〗^2 πœƒ) Γ— 𝑏^2 = 𝑏^2 Hence proved

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Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He provides courses for Mathematics from Class 9 to 12. You can ask questions here.
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