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प्रश्न
Find perpendicular distance from the origin to the line joining the points (cosΘ, sin Θ) and (cosΦ, sin Φ).
उत्तर
The equation of the line joining the points (cosθ, sinθ) and (cos∅, sin∅) is given by
= `y - sin θ = (sin∅ - sinθ)/(cos∅ - cosθ) (x - cosθ)`
= y(cos∅ - cosθ)-sinθ(cos∅ - cosθ) = x(sin∅ - sinθ)-cosθ (sin∅ - sinθ)
= x(sinθ - sin∅)+y(cos∅ - cosθ) + cosθ sin∅ - cosθ sinθ - sinθ cos∅ + sinθ cosθ = 0
= x(sinθ - sin∅)+y(cos∅ - cosθ) + sin (∅ - θ) = 0
= Ax + By + C = 0, where A = sin θ - sin∅, B = cos∅ - cosθ, and C = sin (∅ - θ)
It is known that the perpendicular distance (d) of a line Ax + By + C = 0 from a point (x1, y1) is given by
`d = |Ax_1 + By_1 + C|/sqrt(A^2 + B^2)`
Therefore, perpendicular distance (d) of the given line from the point (x1, y1) = (0, 0) is
`d = |(sinθ - sin∅)(0) + (cos∅ - cosθ)(0) + sin(∅ - θ)|/sqrt((sinθ - sin∅)^2 + (cos∅ - cosθ)^2`
= `|sin (∅ - θ)|/sqrt (sin^2θ + sin^2∅ - 2sinθ sin∅ + cos^2∅ + cos^2θ - 2cos∅ cosθ)`
= `|sin (∅ - θ)|/sqrt ((sin^2θ + cos^2θ) - (sin^2∅ cos^2∅) -2(sinθ - sin∅ + cosθ cos∅)`
= `|sin (∅ - θ)|/sqrt(1 + 1 - 2(cos (∅ - θ)))`
= `|sin (∅ - θ)|/sqrt(2(1 - cos (∅ - θ))`
= `|sin (∅ - θ)|/sqrt(2(2sin^2 ((∅ - θ)/2))`
= `|sin (∅ - θ)|/(|2sin((∅ - θ)/2)|)`
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