Answer:
E. (-3, -4)
Step-by-step explanation:
The given equation is an equation for a circle in standard form. The general form of that equation is ...
(x -h)² +(y -k)² = r²
Comparing the equation you are given with this standard form, you see that ...
(h, k) = (3, -4) . . . . center of the circle
r = 6 . . . . . . . . . . . radius of the circle
This means your circle is centered at (3, -4) and has a radius of 6. A point will be on the circle if it lies a distance of 6 units from the center.
Often, we can look at the differences of coordinates of two points and make a good guess as to their distance apart. So, we can start by computing the differences between your point coordinates and the center of the circle.
A. (9, -2) -(3, -4) = (9 -3, -2 +4) = (6, 2) . . . . farther away than 6 units
B. (0, 11) -(3, -4) = (0 -3, 11 +4) = (-3, 15) . . . . farther away than 6 units
C. (3, 10) -(3, -4) = (3 -3, 10 +4) = (0, 14) . . . . farther away than 6 units
D. (-9, 4) -(3, -4) = (-9 -3, 4 +4) = (-12, 8) . . . . farther away than 6 units
E. (-3, -4) -(3, -4) = (-3 -3, -4 +4) = (-6, 0) . . . . exactly 6 units away
Point E(-3, -4) lies on the circle.
_____
Additional comment
The formula for the distance between two points is based on the Pythagorean theorem. It tells you the distance is ...
d = √((x2 -x1)² +(y2 -y1)²)
So, for some difference between points of (dx, dy), this is ...
d = √(dx² +dy²)
Of course, when a value is squared, the result is positive, and the original sign is of no consequence.
The hypotenuse of a right triangle is longer than the longest side, but shorter than the sum of the sides. Than means the resulting distance represented by (dx, dy) will always lie between the maximum of the magnitudes of dx and dy, and their sum.
In the above list, this means we know the distance from the circle center to the point can be estimated to be ...
A: 6 to 8
B: 15 to 18
C: exactly 14
D: 12 to 20
E: exactly 6 . . . . the distance the question is asking for
