First slide

Definition of a circle

Question

The range of g so that we have always a chord of contact of
tangents drawn from a real point $(a, a)$ to the circle
$x^{2} + y^{2} + 2 g x + 4 y + 2 = 0,$ is

Moderate

A

$(- 3, 0)$
B

$(- 4, 1)$
C

$(- 4, 0)$
D

none of these

Solution

For a real chord of contact of tangents drawn from
$(α, α)$ to the circle $x^{2} + y^{2} + 2 g x + 4 y + 2 = 0$ the point $(α, α)$
must lie outside the circle.
$\begin{array}{ll} ∴ α^{2} + α^{2} + 2 g α + 4 α + 2 > 0 \\ \Rightarrow α^{2} + g α + (2 α + 1) > 0 \\ \Rightarrow α^{2} + (g + 2) α + 1 > 0 \end{array}$
Since a assumes real values. Therefore,
$\begin{array}{l} (g + 2)^{2} - 4 > 0 \\ \Rightarrow g^{2} + 4 g > 0 \Rightarrow - 4 < g < 0 \Rightarrow g \in (- 4, 0) . \end{array}$

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