Q.

Let  S1 &S2 be the two foci of an ellipse and P be a point of the ellipse such that  S1PS2 is a right angle. Area & perimeter of triangle S1PS2 be respectively 30 sq.units & 30 units. 
Let the Length of semi-major axis of the ellipse is  'p' units and Distance between the foci is 'q'  units and Eccentricity of the ellipse is  'r'. Then the value of  [qp+r] is ____, where [.] denotes represents Greatest Integer Function.

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answer is 5.

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Detailed Solution

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As given  S1P+S2P+S1S2=30  &12×S1P×S2P=30,  where  S1P+S2P=2a  &S1S2=2ae.
 Also  (S1P)2+(S2P)2=(S1S2)2
Hence  (S1P+S2P)22×S1P×S2P=(S1S2)2

4a2120=4a2e2&ae+a=152a=17&2ae=13e=1317

p=17/2, q=13, r=13/17

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