1.

Consider an ellipse 4x^(2)=52-13y^(2) and a variable point P on the line y+2x = 12 such that angle F_(1) P F_(2) is maximum where F_(1) and F_(2) are the foci of the given ellipse then ((F_(2)P)/(F_(1)P))^(2) is equal to (where F_(1)lies on positives x-axis)

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Solution :`ANGLE F_(1)P F_(2)`is maximum if P is POINT of tangency of circle through `F_(1),F_(2)&P F_(2)on y+2x =12`

`angleAPF_(1)=angle AF_(2)PimpliesDeltaAPF_(1)~DeltaAF_(2)P`
`implies(F_(1)P)/(F_(2)P)=(AP)/(AF_(2)) & AP^(2)=AF_(1)AF_(2)`
`((PF_(2))/(PF_(1)))^(2)=(AF_(2))/(AF_(1))=3`


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