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If the point \( (2 \alpha, 3 \alpha) \) lies inside the ellipse \( \frac{x^{2}}{4}+\frac{y^{2}}{9}=1 \), then complete set of values of \( \alpha \) is(A) \( \alpha \in(-\infty,-1) \cup(1, \infty) \) (B) \( \alpha \in(-1,1) \)(C) \( \alpha \in(-1,3) \)(D) \( \alpha \in\left(\frac{-1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right) \) |
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Answer» Option (D) is correct. Given that point (2α, 3α) lies inside the ellipse \(\frac{x^2}{4} + \frac{y^2}{9} = 1.\) Therefore, \(\frac{(2 \alpha)^2}{4} + \frac{(3 \alpha)^2}{9} < 1\) \(\Rightarrow\) α2 + α2 < 1 \(\Rightarrow\) 2α2 < 1 \(\Rightarrow\) α2 < \(\frac{1}{2}\) \(\Rightarrow\) α ∈ \(\left(-\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}} \right)\) |
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