Question

The range of values of r for which the point $\left( - 5 + \frac{r}{\sqrt{2}},–3 + \frac{r}{\sqrt{2}} \right)$ is an interior point of the major segment of the circle x² + y² = 16, cut off by the line x + y = 2, is –

• A. (–,$5\sqrt{2}$)
• 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$\sqrt{2}$– $\sqrt{14}$, $5\sqrt{2}$)

Answer 2

The given point is an interior point.

Ž $\left( –5 + \frac{r}{\sqrt{2}} \right)^{2}$+$\left( - 3 + \frac{r}{\sqrt{2}} \right)^{2}$ – 16 < 0

Ž r² –8$\sqrt{2}$r + 18 < 0 Ž 4$\sqrt{2}$–$\sqrt{14}$< r < 4$\sqrt{2}$+$\sqrt{14}$

The point is on the major segment

Ž The centre and the point are on the same side of the line

x + y = 2

Ž –5 + $\frac{r}{\sqrt{2}}$ – 3 + $\frac{r}{\sqrt{2}}$–2 < 0

Ž r < 5$\sqrt{2}$. So, $4\sqrt{2}$– $\sqrt{14}$ < r < $5\sqrt{2}$.