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Q.

Let $S_{1} & S_{2}$ be the two foci of an ellipse and P be a point of the ellipse such that $∠ S_{1} P S_{2}$ is a right angle. Area & perimeter of triangle $S_{1} P S_{2}$ 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 $[q - p + r]$ is ____, where $[.]$ denotes represents Greatest Integer Function.

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Detailed Solution

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As given $S_{1} P + S_{2} P + S_{1} S_{2} = 30 & \frac{1}{2} \times S_{1} P \times S_{2} P = 30,$ where $S_{1} P + S_{2} P = 2 a & S_{1} S_{2} = 2 a e .$
Also ${(S_{1} P)}^{2} + {(S_{2} P)}^{2} = {(S_{1} S_{2})}^{2}$
Hence ${(S_{1} P + S_{2} P)}^{2} - 2 \times S_{1} P \times S_{2} P = {(S_{1} S_{2})}^{2}$

$\Rightarrow 4 a^{2} - 120 = 4 a^{2} e^{2} & a e + a = 15 \Rightarrow 2 a = 17 & 2 a e = 13 \Rightarrow e = \frac{13}{17}$

$p = 17 / 2, q = 13, r = 13 / 17$

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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 [q−p+r] is ____, where [.] denotes represents Greatest Integer Function.