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Q)

State whether the following statement is True or False : The point $(1,2)$ lies inside the circle $x^2+y^2-2x+6y+1=0$.

$\begin{array}{1 1}\text{True}\\\text{False}\end{array}$

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A)
Toolbox:
• If the distance between the centre and a point $P(x,y)$ is < 0 then the point lies inside the circle.
• Distance between two points $A(x_1,y_1),B(x_2,y_2)$ is $\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$
Answer : False
The equation of given circle is $x^2+y^2-2x+6y+1=0$
The given point $P(1,2)$
The coordinates of the centre of the circle is $C(1,-3)$
$\therefore CP=\sqrt{(1-1)^2+(2+3)^2}$
$\Rightarrow 5$
Since $5>0$,the point lies outside the circle.
Hence the given statement is False.