Question

A circle drawn with origin as the centre passes through the point $A\left(\frac{13}{2},0\right)$. Which of the following points does not lie inside the circle?

(a) $\left(\frac{-3}{4},1\right)$
(b) $\left(2,\frac{7}{3}\right)$
(c) $\left(5,\frac{-1}{3}\right)$
(d) $\left(-6,\frac{5}{2}\right)$

Answer 1

(d) $\left(-6,\frac{5}{2}\right)$
Let the given point be $A\left(\frac{13}{2},0\right)$ and the origin (0, 0). Then,
the radius of the circle,
$OA=\sqrt{\left(0-\frac{13}{2}\right)+{\left(0-0\right)}^2}$
$$\begin{gathered} \Rightarrow OA=\sqrt{{\left(\frac{-13}{2}\right)}^2+{\left(0\right)}^2} \\ \Rightarrow OA=\sqrt{\frac{169}{4}} \\ \Rightarrow OA=\frac{13}{2}\text{units} \end{gathered}$$
Now, the distance of the point $\left(\frac{-3}{4},1\right)$ from the origin is
$$\begin{gathered} \sqrt{\left(0+\frac{3}{4}\right)+{\left(0-1\right)}^2} \\ =\sqrt{{\left(\frac{3}{4}\right)}^2+{\left(-1\right)}^2} \\ =\sqrt{\frac{9}{16}+1} \\ =\sqrt{\frac{25}{16}} \\ =\frac{5}{4}\text{units, which lies inside the circle.} \end{gathered}$$
Now, the distance of the point $\left(2,\frac{7}{3}\right)$ from the origin is
$$\begin{gathered} \sqrt{{\left(0-2\right)}^2+{\left(0-\frac{7}{3}\right)}^2} \\ =\sqrt{{\left(-2\right)}^2+{\left(\frac{-7}{3}\right)}^2} \\ =\sqrt{4+\frac{49}{9}} \\ =\sqrt{\frac{85}{9}} \\ =\frac{\sqrt{85}}{3}\text{units, which lies inside the circle.} \end{gathered}$$
Now, the distance of the point $\left(5,\frac{-1}{2}\right)$ from the origin is
$$\begin{gathered} \sqrt{{\left(0-5\right)}^2+{\left(0+\frac{1}{2}\right)}^2} \\ =\sqrt{{\left(-5\right)}^2+{\left(\frac{1}{2}\right)}^2} \\ =\sqrt{25+\frac{1}{4}} \\ =\sqrt{\frac{101}{4}} \\ =\frac{\sqrt{101}}{2}\text{units, which lies inside the circle}. \end{gathered}$$
Now, the distance of the point $\left(-6,\frac{5}{2}\right)$ from the origin is
$$\begin{gathered} \sqrt{{\left(0+6\right)}^2+{\left(0-\frac{5}{2}\right)}^2} \\ =\sqrt{{\left(6\right)}^2+{\left(\frac{-5}{2}\right)}^2} \\ =\sqrt{36+\frac{25}{4}} \\ =\sqrt{\frac{169}{4}} \\ =\frac{13}{2}\text{units=radius OA} \end{gathered}$$
Hence, it does not lie inside the circle.