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

$\begin {array} {1 1} (A)\;\text{ lies inside the circle} & \quad (B)\;\text{ lies outside the circle} \\ (C)\;\text{ lies on the circle} & \quad (D)\;\text{ none of the above} \end {array}$

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A)
Toolbox:
• If the distance between the point through which the circle passes say $(x_1, y_1)$ is less than the radius, then the point lies inside the circle.
• If it is greater then it lies outside the circle.
• If it is equal, then it lies on the circle.
Step 1 :
The equation of the given circle is $x^2+y^2=25$
This can be written as
$(x-0)^2+(y-0)^2=5^2$-------(1)
which is of the form
$(x-h)^2+(y-k)^2=r^2$--------(2)
Comparing both the equations we get,
$h=0, r=0$ and $r=5$
Hence the centre is (0,0) and radius = 5.
$\therefore$ The distance between the point (-2.5, 3.5) and centre (0,0) is
$d = \sqrt{(-2.5-0)^2+(3.5-0)^2}$
$= \sqrt{6.25+12.25}$
$= \sqrt{18.5}$
\$ = 4.3
This distance is lesser than 5
That is less than the radius
Hence the point lies inside the circle.