First slide

Definition of a circle

Question

The set of values of $' a'$ for which the point $(a - 1, a + 1)$ lies outside the circle $X^{2} + Y^{2} = 8$ and inside the circle $x^{2} + y^{2} - 12 x + 12 y - 62 = 0$ , is

Moderate

A

$(- \infty, - \sqrt{3}) \cup (\sqrt{3}, \infty)$
B

$(- 3 \sqrt{2}, 3 \sqrt{2})$
C

$(- 3 \sqrt{2}, - \sqrt{3}) \cup (\sqrt{3}, 3 \sqrt{2})$
D

none of these

Solution

It is given that the point $(a - 1, a + 1)$ lies outside the circle $x^{2} + y^{2} = 8$ and inside the circle $x^{2} + y^{2} - 12 x + 12 y - 62 = 0 .$ Therefore,
$(a - 1)^{2} + (a + 1)^{2} - 8 > 0$
and, $(a - 1)^{2} + (a + 1)^{2} - 12 (a - 1) + 12 (a + 1) - 62 < 0$
$\Rightarrow 2 a^{2} - 6 > 0$ and $2 a^{2} - 36 < 0$
$\Rightarrow a^{2} - 3 > 0$ and $a^{2} - 18 < 0$
$\Rightarrow a \in (- \infty, - \sqrt{3}) \cup (\sqrt{3}, \infty)$ and $a \in (- 3 \sqrt{2}, 3 \sqrt{2})$
$\Rightarrow a \in (- 3 \sqrt{2}, - \sqrt{3}) \cup (\sqrt{3}, 3 \sqrt{2})$

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