The line $x = t$ meets the ellipse $x^{2} + y^{2} = 1$ in real and distinct points if and only if:

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Solution

Let's analyze the condition
Solving the given line and the ellipse, we get
$t^{2} + 9 y^{2} = 1 \Rightarrow y^{2} = 9 (1 - t^{2})$
Which gives real and distinct values of $y$ , if $1 - t^{2} > 0$
$\Rightarrow$ $t \in (- 1, 1)$
Therefore, the line $x = t$ meets the ellipse $x^{2} + y^{2} = 1$ in real and distinct points when $|t| < 1 .$

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