First slide

Definition of a circle

Question

A region in the x-y plane is bounded by the curve $y = \sqrt{25 - x^{2}}$ and the line y =0 If the point (a, a+ 1) lies in the interior of the region then,

Moderate

A

$a \in(-4,3)$
B

$a \in (- \infty, - 1) \in (3, \infty)$
C

$a \in (- 1, 3)$
D

none of these

Solution

$y = \sqrt{25 - x^{2}} and y = 0$ bound the semicircle above the x-axis. Therefore,
$a + 1 > 0$ …………(i)
and $a^{2} + (a + 1)^{2} - 25 < 0 or 2 a^{2} + 2 a - 24 < 0$
or $a^{2} + a - 12 < 0$
or $- 4 < a < 3$ …….(ii)
From (i) and (ii),
$- 1 < a < 3$

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