Consider the parabola represented by the parametric equations $x = t^{2} - 2 t + 2; y = t^{2} + 2 t + 2$ . Then which of the following is/are true?

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Vertex of the parabola is (2, 2)

Auxiliary circle of the parabola is $x + y = 4$

Director circle of the parabola is $x + y = 6$

Focus of the parabola is (3, 3)

answer is A, B, D.

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Detailed Solution

$x = t^{2} - 2 t + 2; y = t^{2} + 2 t + 2$
$x + y = 2 (t^{2} + 2)$ and $y - x = 4 t$
$\frac{x + y}{2} = \frac{{(y - x)}^{2}}{16} + 2 \Rightarrow {(y - x)}^{2} = 8 (x + y - 4)$
${(\frac{y - x}{\sqrt{2}})}^{2} = 4 \sqrt{2} (\frac{x + y - 4}{\sqrt{2}})$
This is parabola for which $y = x$ is axis, $x + y = 4$ is tangent at vertex and length of latusrectum is $4 \sqrt{2}$