Question

Given the circle with the equation (x - 3)2 + y2 = 49, determine the location of each point with respect to the graph of the circle. In your final answer, state whether each point is on the interior, exterior, or circumference of the circle. Include your calculations as proof of each point’s location.
A. (-1, 1)
B. (10, 0)
C. (4, -8)

Answer 1

To find out if a point is inside, on, or outside a circle, we need to substitute the ordered pair into the equation of the circle:
(x-xc)^2+(y-yc)^2=r^2
where (xc,yc) is the centre of the circle, and r=radius of the circle.

If the left-hand side [(x-xc)^2+(y-yc)^2] is less than r^2, then point (x,y) is INSIDE the circle.  If the left-hand side is equal to r^2, the point is ON the circle.
Finally, if the left-hand side is greater than r^2, the point is OUTSIDE the circle.

For the given problem, we have xc=3, yc=0, or centre at (3,0), r=sqrt(49)=7
(x-xc)^2+(y-yc)^2=r^2 => (x-3)^2+y^2=7^2

A. (-1,1), 
(x-3)^2+y^2=7^2 => (-1-3)^2+1^2=16+1=17 <49  [inside circle]

B. (10,0)
(x-3)^2+y^2=7^2 => (10-3)^2+0^2=49+0=49  [on circle]

C. (4,-8)
(x-3)^2+y^2=7^2 => (4-3)^2+(-8)^2=1+64=65 > 49  [outside circle]