Question

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

Answer 1

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.