Consider a circle whose radius is 2 and centre is origin. Tangents are drawn from a point $P (α, 2)$ which touches the circle at $A (a_{1}, a_{2})$ and $B (b_{1}, b_{2}) .$ Another pair of tangents are drawn from the point $Q (β, - 2)$ which touches the circle at $A (a_{1}, a_{2})$ and $C (c_{1}, c_{2}) .$ Then which of the following is/are CORRECT ? $(α, β > 0)$

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$b_{1} + b_{2} + c_{1} + c_{2} = 0$

$α β = 4$

If $α = 4,$ then $4 a_{2} + 3 a_{1} = 0$

If $α = β,$ then $a_{2} = 0$

answer is A, B, C, D.

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Detailed Solution

$B (0, 2)$ and $C (0, - 2)$ . Slope of $P Q = \frac{4}{α - β}$
Equation of $P Q; y - 2 = \frac{4}{α - β} (x - α)$
$y (α - β) - 2 (α - β) = 4 x - 4 α 4 x - y (α - β) - 2 (α + β) = 0 ∵ it touches x^{2} + y^{2} = 4 ∴ \frac{| 0 - 0 - 2 (α + β) |}{\sqrt{16 + {(α - β)}^{2}}} = 2$
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$\Rightarrow a β = 4 \Rightarrow i f α = 4 s l o p e o f P Q = \frac{4}{3}$
$∴ s l o p e o f O A = - \frac{3}{4} \Rightarrow \frac{a_{2}}{a_{1}} = - \frac{3}{4}$