Points P(-3, 2), Q(9, 10) and R(, 4) lie on a circle C with PR as its diameter. The tangents to C at the points Q and R intersect at the point S. If S lies on the line 2 x - k y = 1 , then k is equal to ______ . [2023]

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Solution

Q.
Points P(-3, 2), Q(9, 10) and R( $α$ , 4) lie on a circle C with PR as its diameter. The tangents to C at the points Q and R intersect at the point S. If S lies on the line $2 x - k y = 1$ , then $k$ is equal to ______ . [2023]

Ans.
(3)

$Since P Q ⊥ Q R, so m_{P Q} \cdot m_{Q R} = - 1$

$\Rightarrow \frac{10 - 2}{9 + 3} \times \frac{10 - 4}{9 - α} = - 1 \Rightarrow α = 13$

$Since Q O ⊥ Q S, so m_{O Q} \cdot m_{Q S} = - 1 \Rightarrow m_{Q S} = - \frac{4}{7}$

$Equation of Q S : y - 10 = - \frac{4}{7} (x - 9) \Rightarrow 4 x + 7 y = 106 (i)$

$Since O R ⊥ S R$

$m_{O R} \cdot m_{R S} = - 1$

$\Rightarrow m_{R S} = - 8$

$Equation of R S : y - 4 = - 8 (x - 13)$

$\Rightarrow 8 x + y = 108 (ii)$

$Solving (i) and (ii)$

$x = \frac{25}{2}, y = 8$

$Since S (x, y) lies on the line 2 x - k y = 1$

$25 - 8 k = 1 \Rightarrow 8 k = 24 \Rightarrow k = 3$

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