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Does the point (1,√3) lie inside outside or on the circle with a center at the origin containing the point (3,0)? Question 2 options: Inside On Outside

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Answer:

Point
\left(1,√(3)\right) lies inside the circle.

Explanation:

To determine the point which lies outside the circle, calculate the distance between the points and compare that value with radius of circle.

If d<r point lies inside the circle, d>r point lies outside circle and d=r point is on the circle.

Let O be the center of origin of the circle. So,
O=\left ( 0,0 \right ).

Also given that circle contain point
\left ( 3,0 \right ). So,
x=3,y=0

Now equation of circle having center (h,k) is given by the equation,


\left(x-h\right)^2+\left(y-k\right)^2=r^2


h=0,k=0

Substituting the value,


\left(3-0\right)^2+\left(0-0\right)^2=r^2


9=r^2


r=\pm 3

Now calculate the distance between point
\left(3,0\right) and
\left (1,√(3)\right).

Distance formula between points
\left(x_(1),y_(1)\right) and
\left(x_(2),y_(2)\right)is given as,


d=√(\left(x_2-x_1\right)^2+\left(y_2-y_1\right)^2)

Now,
x_(1)=1,y_(1)=√(3),x_(2)=3,y_(2)=0


\therefore d=\sqrt{\left(3-1\right)^2+\left(0-√(3)\right)^2}

Simplifying,


\therefore d=\sqrt{\left(2\right)^2+\left(-√(3)\right)^2}


\therefore d=√(4+3)


\therefore d=√(7)


\therefore d=2.64

Since r = 3 and d = 2.64. That is, d < r.

So point
\left (1,√(3)\right) lies inside the circle.

User Eleonor
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