Example 19 - A rod AB of length 15 cm rests in between two axes - Examples

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  1. Chapter 11 Class 11 Conic Sections
  2. Serial order wise
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Example, 19 A rod AB of length 15 cm rests in between two coordinate axes in such a way that the end point A lies on x-axis and end point B lies on y-axis. A point P(x, y) is taken on the rod in such a way that AP = 6 cm. Show that the locus of P is an ellipse. Given AB = 15 cm & AP = 6 cm PB = AB – AP PB = 15 – 6 PB = 9cm Drawing PQ ⊥ BO and PR ⊥ OA Hence, PQ = x & PR = y Let ∠ PAR = θ Now, PQ & AO are parallel lines (As both are perpendicular to y-axis) & BA is the transversal So ∠ BPQ = ∠ PAR = θ Now we know that, ﷐ sin2﷮𝜃﷯ + ﷐cos2﷮𝜃﷯ = 1 Putting ﷐ 𝑠𝑖𝑛﷮𝜃﷯ = ﷐𝑦﷮6﷯ and ﷐𝑐𝑜𝑠﷮𝜃﷯ = ﷐𝑥﷮9﷯ ﷐﷐﷐𝑦﷮6﷯﷯﷮2﷯ + ﷐﷐﷐𝑥﷮9﷯﷯﷮2﷯ = 1 ﷐𝑦2﷮36﷯ + ﷐𝑥2﷮81﷯ = 1 ﷐𝒙𝟐﷮𝟖𝟏﷯ + ﷐𝒚𝟐﷮𝟑𝟔﷯ = 1 Hence it satisfies the equation of ellipse ﷐﷐𝑥﷮2﷯﷮﷐𝑎﷮2﷯﷯ + ﷐﷐𝑦﷮2﷯﷮﷐𝑏﷮2﷯﷯ = 1 Thus locus of P is ellipse Hence proved

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