There are two sets of parallel lines, their equations being xcosα+ysinα=p and ycosα−xcosα=p;p=1,2,3,....,n and α ∈ (0,π/2). If the number of rectangles formed by these two sets of lines is 225, then find the value of n
Answers
Given : There are two sets of parallel lines, their equations being xcosα+ysinα=p and ycosα−xcosα=p;p=1,2,3,....,n and α ∈ (0,π/2).
the number of rectangles formed by these two sets of lines is 225,
To Find : the value of α and n
Solution:
xcosα+ysinα=p
=> ysinα = -xcosα + p
=> ysinα = -xcosα/sinα + p/sinα
=> y = - xcotα + p/sinα
Slope = - cotα
ycosα−xcosα=p
=>y = x + p/cosα
Slope = 1
=> - cotα * 1 = - 1
=> cotα = 1
=> α = π/4
to have a rectangle we need to select 2 lines from each set
ⁿC₂.ⁿC₂ = 225
=> ⁿC₂.ⁿC₂ = 15 * 15
=> ⁿC₂ = 15
=> n(n -1)/2 = 15
=> n(n - 1) = 30
=> n² - n - 30 = 0
=> (n - 6)(n + 5) = 0
=> n = 6
value of n = 6
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