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The straight line (x)/(4)+(y)/(3) =1 intersects the ellipse (x^(2))/(16)+(y^(2))/(9) =1 at two points A and B, there is a point P on this ellipse such that the area of DeltaPAB is equal to 6(sqrt(2)-1). Then the number of such points (P) is/are

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Solution :Let the perpendicular DISTANCE of P from the line be H,

`:. (1)/(2) xx h xx 5 = 6 (sqrt(2)-1)`
`:. h = (12)/(5) (sqrt(2)-1)`
Also tangent parallel to the given line is `4y + 3X = 12 sqrt(2)`
Its distance from the line `4y +3x = 12` is `h =(12)/(5) (sqrt(2)-1)` HENCE there are three points as shown in the figure.


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