First slide

Definition of a circle

Question

The set of values of a for which the point $(2 a, a + 1)$ is an interior point of the larger segment of the circle $x^{2} + y^{2} - 2 x - 2 y - 8 = 0$ made by the chord $x - y + 1 = 0,$ is

Moderate

A

$(- 1, 9 / 5)$
B

$(0, 9 / 5)$
C

$(0, \infty)$
D

none of these

Solution

The point $(2 a, a + 1)$ will be an interior point of the larger segment of the circle $x^{2} + y^{2} - 2 x - 2 y - 8 = 0$ ,
if (i) the point $(2 a, a + 1)$ is an interior point
and, (ii) the point $(2 a, a + 1)$ and the centre $(1, 1)$ are on the same
side of the chord $x - y + 1 = 0 .$
and,
$(2 a - a - 1 + 1) (1 - 1 + 1) > 0$
$\Rightarrow 5 a^{2} - 4 a - 9 < 0$ and $a > 0$
$\Rightarrow (5 a - 9) (a + 1) < 0$ and $a > 0$
$\Rightarrow - 1 < a < \frac{9}{5}$ and $a > 0 \Rightarrow a \in (0, 9 / 5)$

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