Let P Q be a diameter of the circle x 2 + y 2 = 9 . If α and β are the lengths of the…

Let PQ be a diameter of the circle x2+y2=9. If α and β are the lengths of the perpendiculars from P and Q on the straight line, x+y=2 respectively, then the maximum value of αβ is _______

Solution

 Let P(3cosθ,3sinθ)∴Q(-3cosθ,-3sinθ)
given line x+y-2=0 ∴α=|3cosθ+3sinθ-2|2
β=|-3cosθ-3sinθ-2|2
∴  αβ=(3cosθ+3sinθ-2)·(3cosθ+3sinθ+2)2=9(1+sin2θ)-42
∴ maximum of αβ=7

Asked in: JEE Main 2020 (04 Sep Shift 2)

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