Bunuel
A circle in the XY-plane has its center at the origin. If M is a point on the circle, what is the sum of the squares of the coordinates of M?
(1) The radius of the circle is 5.
(2) The sum of the coordinates of M is 7.
Solution:
Since the circle is centered at the origin, if it has a radius of r, then an equation of the circle is x^2 + y^2 = r^2. Since M is a point on the circle, if M has the coordinates of (a, b), then a^2 + b^2 = r^2.
We need to determine the value of a^2 + b^2. We can see that if we know the value of r, then we can determine the value of a^2 + b^2 since it’s equal to r^2.
Statement One Only:
The radius of the circle is 5.
This means r = 5. Since r = 5, we see that a^2 + b^2 = 5^2 = 25. Statement one alone is sufficient.
Statement Two Only:
The sum of the coordinates of M is 7.
This means a + b = 7. If a = 3 and b = 4, a^2 + b^2 = 9 + 16 = 25. However, if a = 2 and b = 5, a^2 + b^2 = 4 + 25 = 29. Statement two alone is not sufficient.
Answer: A