Does the point (–2.5, 3.5) lie inside,

Question:

Does the point $(-2.5,3.5)$ lie inside, outside or on the circle $x^{2}+y^{2}=25 ?$

Solution:

The equation of the given circle is $x^{2}+y^{2}=25$.

$x^{2}+y^{2}=25$

$\Rightarrow(x-0)^{2}+(y-0)^{2}=5^{2}$, which is of the form $(x-h)^{2}+(y-k)^{2}=r^{2}$, where $h=0, k=0$, and $r=5$

$\therefore$ Centre $=(0,0)$ and radius $=5$

Distance between point $(-2.5,3.5)$ and centre $(0,0)$

$=\sqrt{(-2.5-0)^{2}+(3.5-0)^{2}}$

$=\sqrt{6.25+12.25}$

$=\sqrt{18.5}$

$=4.3$ (approx.) $<5$

Since the distance between point (–2.5, 3.5) and centre (0, 0) of the circle is less than the radius of the circle, point (–2.5, 3.5) lies inside the circle.