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Let the circle ${(x - 1)}^{2} + {(y - 2)}^{2} = 25$ cuts a rectangular hyperbola with transverse axis along $y = x$ at four points $A, B, C$ and $D$ having coordinates $(x_{i}, y_{i}), i = 1, 2, 3, 4$ respectively. O being the center of the hyperbola. Now match the entries from the following two columns
Column–I Column–II
A
$x_{1} + x_{2} + x_{3} + x_{4}$ is equal to
P
2
B
$x_{1}^{2} + x_{2}^{2} + x_{3}^{2} + x_{4}^{2}$ is equal to
Q
4
C
$O A^{2} + O B^{2} + O C^{2} + O D^{2}$ is equal to
R
44
D
$y_{1}^{2} + y_{2}^{2} + y_{3}^{2} + y_{4}^{2}$ is equal to
S
56

T
100

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$A \to P, B \to R, C \to T, D \to S$

$A \to P, B \to Q, C \to R, D \to S$

$A \to P, B \to T, C \to Q, D \to S$

$A \to S, B \to Q, C \to R, D \to P$

answer is C.

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Detailed Solution

$x^{2} + y^{2} - 2 x - 4 y - 20 = 0 \to (1)$

$x y = c^{2} \to (2)$

$P.I of (1) & (2) is P (c t, \frac{c}{t})$

$\Rightarrow c^{2} t^{4} - 2 c t^{3} - 20 t^{2} - 4 c t + c^{2} = 0$

$A) x_{1} + x_{2} + x_{3} + x_{4} = c (t_{1} + t_{2} + t_{3} + t_{4}) = 2$

$B) \sum {x_{1}}^{2} = {(\sum x_{1})}^{2} - 2 \sum x_{1} x_{2} = 44$

$C) \sum (x_{1}^{2} + y_{1}^{2}) = 44 + 56 = 100$

$D) \sum y_{1} = c (\frac{1}{t_{1}} + \frac{1}{t_{2}} + \frac{1}{t_{3}} + \frac{1}{t_{4}}) = 4$

$\sum y_{1} y_{2} = c^{2} \sum \frac{1}{t_{1} t_{2}} = - 20$

$\sum y_{1}^{2} = {(\sum y_{1})}^{2} - 2 \sum y_{1} y_{2} = 56$

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