Question

The range of values of $r$ for which the point $(-5+\frac{r}{\sqrt{2}},-3+\frac{r}{\sqrt{2}})$ is an interior point of the major segment of the circle $x^2+y^2=16$, cut-off by the line $x+y=2$, is

• A. $(-\infty ,5\sqrt{2})$
• B. $(4\sqrt{2}-\sqrt{14},5\sqrt{2})$
• C. $(4\sqrt{2}-\sqrt{14},4\sqrt{2}+\sqrt{14})$
• D. None of these

Answer 1

Answer: (B)

$(4\sqrt{2}-\sqrt{14},5\sqrt{2})$

The given point is an interior point
$\Rightarrow {(-5+\frac{r}{\sqrt{2}})}^2+{(-3+\frac{r}{\sqrt{2}})}^2-16<0$
$\Rightarrow r^2-8\sqrt{2}r+18<0$
$\Rightarrow 4\sqrt{2}-\sqrt{14}<r<4\sqrt{2}+\sqrt{14}$ ..... (i)
The point is on the major segment
$\Rightarrow$ The centre and the point are on the same side of the line $x+y=2$
$\Rightarrow -5+\frac{r}{\sqrt{2}}-3+\frac{r}{\sqrt{2}}-2<0$
$\Rightarrow r<5\sqrt{2}$ ........ (ii)
From Eqs. (i) and (ii). $4\sqrt{2}-\sqrt{14}<r<5\sqrt{2}$